According to literature on brand loyalty, consumers who are loyal to a brand are likely to consistently select the same product. This type of consistency could come from a positive childhood association. To examine brand loyalty among fans of the Chicago Cubs, 365 Cubs fans among patrons of a restaurant located in Wrigleyville were surveyed prior to a game at Wrigley Field, the Cubs' home field. The respondents were classified as "die-hard fans" or "less loyal fans." Of the 134 die-hard fans, 88.1% reported that they had watched or listened to Cubs games when they were children. Among the 231 less loyal fans, 71.0% said that they watched or listened as children. (Let D = p_die-hard − p_{less loyal}.)

(a) Find the numbers of die-hard Cubs fans who watched or listened to games when they were children. Do the same for the less loyal fans. (Round your answers to the nearest whole number.)

(b) Use a one sided significance test to compare the die-hard fans with the less loyal fans with respect to their childhood experiences relative to the team. (Use your rounded values from part (a). Use α = 0.01. Round your z-value to two decimal places and your P-value to four decimal places.)

Conclusion

____Reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.

____Fail to reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.    

____Fail to reject the null hypothesis, there is not significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.

____Reject the null hypothesis, there is not significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.

(c) Express the results with a 95% confidence interval for the difference in proportions. (Round your answers to three decimal places.)

( ______ , ______ )

Thank you
,

A medical researcher wants to compare the pulse rates of smokers and non-smokers. He believes that the pulse rate for smokers and non-smokers is different and wants to test this claim at the 0.02 level of significance. A sample of 46 smokers has a mean pulse rate of 75, and a sample of 47 non-smokers has a mean pulse rate of 73. The population standard deviation of the pulse rates is known to be 7 for smokers and 10 for non-smokers. Let μ1 be the true mean pulse rate for smokers and μ2 be the true mean pulse rate for non-smokers. Step 1 of 4: State the null and alternative hypotheses for the test. Step 2 of 4: Compute the value of the test statistic. Round your answer to two decimal places. Step 3 of 4: Determine the decision rule for rejecting the null hypothesis H0. Round the numerical portion of your answer to two decimal places. Step 4 of 4: Make the decision for the hypothesis test.

A publisher reports that 42% of their readers own a particular make of car. A marketing executive wants to test the claim that the percentage is actually different from the reported percentage. A random sample of 250 found that 35% of the readers owned a particular make of car. Find the value of the test statistic. Round your answer to two decimal places.

The National Student Loan Survey asked the student loan borrowers in their sample about attitudes toward debt. Below are some of the questions they asked, with the percent who responded in a particular way. Assume that the sample size is 1260 for all these questions. Compute a 95% confidence interval for each of the questions, and write a short report about what student loan borrowers think about their debt. (Round your answers to three decimal places.)

(a) "To what extent do you feel burdened by your student loan payments?" 56.9% said they felt burdened.

( _____ , ______ )

  ,

(b) "If you could begin again, taking into account your current experience, what would you borrow?" 55.2% said they would borrow less.

(_____ , ______ )

   

(c) "Since leaving school, my education loans have not caused me more financial hardship than I had anticipated at the time I took out the loans." 33.5% disagreed.

( ____ , _____ )

(d) "Making loan payments is unpleasant but I know that the benefits of education loans are worth it." 58.4% agreed.

( ____ , _____ )

(e) "I am satisfied that the education I invested in with my student loan(s) was worth the investment for career opportunities." 58.6% agreed.

( _____ , _____ )

  , (f) "I am satisfied that the education I invested in with my student loan(s) was worth the investment for personal growth." 70.6% agreed.

( ____ , _____ )

,
Conclusion

____While many feel that loans are a burden and wish they had borrowed less, a minority are satisfied with their education.

____While many feel that loans are a burden and wish they had borrowed less, a majority are satisfied with their education.

____While a minority feel that loans are a burden and wish they had borrowed more, a minority are satisfied with their education.

____While a minority feel that loans are a burden and wish they had borrowed more, a majority are satisfied with their educatio

THANK YOU

To what extent do syntax textbooks, which analyze the structure of sentences, illustrate gender bias? A study of this question sampled sentences from 10 texts. One part of the study examined the use of the words "girl," "boy," "man," and "woman." We will call the first two words juvenile and the last two adult. Is the proportion of female references that are juvenile (girl) equal to the proportion of male references that are juvenile (boy)? Here are data from one of the texts:

(a) Find the proportion of juvenile references for females and its standard error. Do the same for the males. (Round your answers to three decimal places.)

Answers:

(b) Give a 90% confidence interval for the difference. (Do not use rounded values. Round your final answers to three decimal places.)

( ______ , _____ ) Answers

(c) Use a test of significance to examine whether the two proportions are equal. (Use p̂_F − p̂_M. Round your value for z to two decimal places and round your P-value to four decimal places.)

Answers :

State your conclusion.

___There is not sufficient evidence to conclude that the two proportions are different.

___There is sufficient evidence to conclude that the two proportions are different.    

THANK YOU :)

Statistics Question

Data provided below

(To be done with EVIEWS or any other data processor)

d)

e) In general, we can conduct hypothesis tests on a population central location with EViews by performing the (one sample) t-test, the sign test or the Wilcoxon signed ranks test.2 Suppose we would like to know whether there is evidence at the 5% level of significance that the population central location of NAR is larger than 5%. which test(s) offered by EViews would be the most appropriate this time? Explain your answer by considering the conditions required by these tests.

(f) Perform the test you selected in part (e) above with EViews. Do not forget to specify the null and alternative hypotheses, to identify the test statistic, to make a statistical decision based on the p-value, and to draw an appropriate conclusion. If the test relies on normal approximation, also discuss whether this approximation is reasonable this time.

(g) Perform the other tests mentioned in part (e). Again, do not forget to specify the null and alternative hypotheses, to identify the test statistics, to make statistical decisions based on the p-values, and to draw appropriate conclusions. Also, if the tests rely on normal approximation, discuss whether these approximations are reasonable this time.

(h) Compare your answers in parts (f) and (g) to each other. Does it matter in this case whether the population of net returns is normally, or at least symmetrically distributed or not? Explain your answer.

Suppose that you roll a die and your score is the number shown on the die. On the other
hand, suppose that your friend rolls five dice and his score is the number of 6’s shown out of five rollings. Compute the probability
(a) that the two scores are equal.
(b) that your friend’s score is strictly smaller than yours.

. A chemist wishes to detect an impurity in a certain compound that she is making. There is a test that detects an impurity with probability 0.92; however,

This test indicates that an impurity is there when it is not about 5% of the time. The chemist produces compounds with the impurity about 15% of the time. A compound is selected at random from the chemist’s output. The test indicates that an impurity is present. What is the conditional probability that the compound actually has the impurity?

Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body bone mineral content (TBBMC). A highly skilled operator is required to take the measurements. Recently, a new DXA machine was purchased by a research lab, and two operators were trained to take the measurements. TBBMC for eight subjects was measured by both operators. The units are grams (g). A comparison of the means for the two operators provides a check on the training they received and allows us to determine if one of the operators is producing measurements that are consistently higher than the other. Here are the data.

(a) Take the difference between the TBBMC recorded for Operator 1 and the TBBMC for Operator 2. (Use Operator 1 minus Operator 2. Round your answers to four decimal places.)

x=________

s=________

Describe the distribution of these differences using words.

___The distribution is uniform.

___The distribution is left skewed.    

___The sample is too small to make judgments about skewness or symmetry.

___The distribution is Normal.The distribution is right skewed.

(b) Use a significance test to examine the null hypothesis that the two operators have the same mean. Give the test statistic. (Round your answer to three decimal places.)

t = ________

Give the degrees of freedom.__________

Give the P-value. (Round your answer to four decimal places.) ___________

Give your conclusion. (Use the significance level of 5%.)

____We cannot reject H₀ based on this sample.

____We can reject H₀ based on this sample.    

(c) The sample here is rather small, so we may not have much power to detect differences of interest. Use a 95% confidence interval to provide a range of differences that are compatible with these data. (Round your answers to four decimal places.)

(______________ , _____________ )

  ,

(d) The eight subjects used for this comparison were not a random sample. In fact, they were friends of the researchers whose ages and weights were similar to the types of people who would be measured with this DXA machine. Comment on the appropriateness of this procedure for selecting a sample, and discuss any consequences regarding the interpretation of the significance-testing and confidence interval results.

____The subjects from this sample may be representative of future subjects, but the test results and confidence interval are suspect because this is not a random sample.

_____The subjects from this sample, test results, and confidence interval are representative of future subjects.

A study of iron deficiency among infants compared samples of infants following different feeding regimens. One group contained breast-fed infants, while the infants in another group were fed a standard baby formula without any iron supplements. Here are summary results on blood hemoglobin levels at 12 months of age.

(a)

Is there significant evidence that the mean hemoglobin level is higher among breast-fed babies? State H₀ and H_a.H₀:

____H₀: μ_breast-fed > μ_formula;   H_a: μ_breast-fed = μ_formula

_{_____H0}: μ_breast-fed < μ_formula;   H_a: μ_breast-fed = μ_formula  

____H₀: μ_breast-fed ≠ μ_formula;   H_a: μ_breast-fed < μ_formula

____H₀: μ_breast-fed = μ_formula;   H_a: μ_breast-fed > μ_formula

Carry out a t test. Give the P-value. (Use α = 0.01. Use μ_breast-fed − μ_formula. Round your value for t to three decimal places, and round your P-value to four decimal places.)

What is your conclusion?

____Reject the null hypothesis. There is significant evidence that the mean hemoglobin level is higher among breast-fed babies.

____Reject the null hypothesis. There is not significant evidence that the mean hemoglobin level is higher among breast-fed babies.   

____ Fail to reject the null hypothesis. There is not significant evidence that the mean hemoglobin level is higher among breast-fed babies.

____Fail to reject the null hypothesis. There is significant evidence that the mean hemoglobin level is higher among breast-fed babies.

(b)

Give a 95% confidence interval for the mean difference in hemoglobin level between the two populations of infants. (Round your answers to three decimal places.)

________, _________ Answers

__C)

State the assumptions that your procedures in (a) and (b) require in order to be valid.

____We need two dependent SRSs from normal populations.

____We need sample sizes greater than 40.  

____ We need two independent SRSs from normal populations.

_____We need the data to be from a skewed distribution.

Prove:
(?) ???=Σ ( ?−?̅  )² =Σ ?²−[(Σ ?²)/n] (Sum Squares X)

(?) ???=Σ( ?−?̅ )²=Σ ?²−[(Σ ?²)/?] (Sum Squares Y)

(?) ??? =Σ( ?−?̅ ) ( ?−?̅ )=Σ ??−[(Σ?)(Σ?)?] (Sum Products X,Y)

HOMEWORK 1

This assignment is designed to illustrate how a software package such as Microsoft Excel supplemented by an add-in such as PHStat can enable one to calculate minimum sample sizes necessary in order to construct confidence intervals for both population means and proportions and to construct these types of confidence intervals. You should use PHStat in order to accomplish all parts of this assignment. You should not only find the required information, but you should explain the meanings of your results for each problem and part of each problem in the context of the problem. You also should provide business implications of the results at which you arrive for one part of either problems two and three and for problem five.

Scenario of the Problem:

1. You have been asked by a certain political party to study the mean age of the supporters of a certain candidate who is running for public office in an upcoming election. A random sample of those who have demonstrated their support for the candidate will be chosen in order to accomplish the desired study. In order to provide estimates of the population mean age of the supporters of this candidate, what minimum sample sizes will be necessary under the following conditions?                          

1. The estimate desired will need to be computed with 98% confidence to within ±2 years when it is felt that the population standard deviation in the ages of the supporters of the candidate is 7.5 years.            
2. The estimate desired will now need to be computed with 95% confidence to within ±2 years when the population standard deviation is 7.5 years.                                                                          
3. The estimate desired will now need to be computed with 98% confidence to within ±2 years when the population standard deviation is 6 years.                                                                              
4. The estimate desired will now need to be computed with 98% confidence to within ±3 years when the population standard deviation is 7.5 years.                                                                          

In your memo, be sure to comment on the differences found in the calculation of the minimum sample sizes in the various parts of the above problem. Explain why differences in your answers exist. In doing so, make all comparisons relative to the answer found in the first part of the problem.                                                                                        

2. You now need to construct a confidence interval for the mean age of the supporters of the candidate. You select a random sample of 80 identified supporters of the candidate. You find that their mean age is 44.57 years. You believe that the population standard deviation of the ages of the supporters of the candidate is 7.5 years. Construct both 98% and 95% confidence intervals for the mean age of the supporters of the candidate. In your explanation, comment upon the effect of the change in confidence level on the width of your interval.

3. You no longer believe that the population standard deviation in the ages of the supporters of the candidate is a known quantity. You therefore will use the sample standard deviation of the ages of the supporters as an estimate of this unknown population standard deviation. You collect data from a random sample of supporters of the candidate. The data identifies the ages of a sample of the supporters of the candidate. This data is shown in appendix one below. Construct both 98% and 95% confidence intervals for the mean age of the supporters of the candidate for this situation. At each confidence level, comment upon the change in the results of this problem from the results of the previous problem.

            Appendix One: (Age of Supporters)

            40        32        60        58        22        28        66        70        71        55        59        58        62        44        89        48        56        33        46            39        39        44        32        48        49        50        51        18        28        23        34        54        28        76        35        77        38        21            59        51        54        38        45        39        19        90        37        46        22        26        27        39        30        45        27       

4. You also need to estimate the population proportion of supporters of the candidate that are usually loyal supporters of the political party that this candidate represents based upon their attesting to this fact and their previous voting record. What minimum sample sizes will be necessary in order to estimate the desired population proportion under the following conditions?

1. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is thought to equal 80%.
2. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is unknown.
3. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is thought to equal 95%.

Comment on the changes in the minimum sample sizes you have computed based upon the changes in the information given in the three parts of this problem.

5. You now need to estimate with 98% confidence the population proportion of supporters of the candidate that describes itself as loyal to the political party represented by the candidate. You randomly sample the population of supporters of the candidate and ascertain whether each one has been a loyal party supporter. The results of that sampling process are shown in appendix two below. Using this information, construct the required confidence interval.

Appendix Two: (Loyal Party Supporter? (Y = yes, N = no))

Y         Y         Y         Y         N         N         Y         Y         Y         Y         N

Y         N         Y         Y         Y         Y         Y         Y         Y         N         Y

N         Y         Y         Y         Y         Y         Y         Y         Y         Y         Y

Y         Y         N         N         N         Y         Y         Y         Y         Y         Y        

Y         Y         Y         Y         Y         Y         N         Y         N         N         Y        

N         Y         Y         Y         Y         Y         Y         Y         Y         N         Y

Y         Y         Y         N         Y         Y         Y         Y         Y         Y         N

N         Y         Y         Y         Y         Y         Y         Y         Y         Y         Y        

HOMEWORK 1

This assignment is designed to illustrate how a software package such as Microsoft Excel supplemented by an add-in such as PHStat can enable one to calculate minimum sample sizes necessary in order to construct confidence intervals for both population means and proportions and to construct these types of confidence intervals. You should use PHStat in order to accomplish all parts of this assignment. You should not only find the required information, but you should explain the meanings of your results for each problem and part of each problem in the context of the problem. You also should provide business implications of the results at which you arrive for one part of either problems two and three and for problem five.

Scenario of the Problem:

1. You have been asked by a certain political party to study the mean age of the supporters of a certain candidate who is running for public office in an upcoming election. A random sample of those who have demonstrated their support for the candidate will be chosen in order to accomplish the desired study. In order to provide estimates of the population mean age of the supporters of this candidate, what minimum sample sizes will be necessary under the following conditions?                          

1. The estimate desired will need to be computed with 98% confidence to within ±2 years when it is felt that the population standard deviation in the ages of the supporters of the candidate is 7.5 years.            
2. The estimate desired will now need to be computed with 95% confidence to within ±2 years when the population standard deviation is 7.5 years.                                                                          
3. The estimate desired will now need to be computed with 98% confidence to within ±2 years when the population standard deviation is 6 years.                                                                              
4. The estimate desired will now need to be computed with 98% confidence to within ±3 years when the population standard deviation is 7.5 years.                                                                          

In your memo, be sure to comment on the differences found in the calculation of the minimum sample sizes in the various parts of the above problem. Explain why differences in your answers exist. In doing so, make all comparisons relative to the answer found in the first part of the problem.                                                                                        

2. You now need to construct a confidence interval for the mean age of the supporters of the candidate. You select a random sample of 80 identified supporters of the candidate. You find that their mean age is 44.57 years. You believe that the population standard deviation of the ages of the supporters of the candidate is 7.5 years. Construct both 98% and 95% confidence intervals for the mean age of the supporters of the candidate. In your explanation, comment upon the effect of the change in confidence level on the width of your interval.

3. You no longer believe that the population standard deviation in the ages of the supporters of the candidate is a known quantity. You therefore will use the sample standard deviation of the ages of the supporters as an estimate of this unknown population standard deviation. You collect data from a random sample of supporters of the candidate. The data identifies the ages of a sample of the supporters of the candidate. This data is shown in appendix one below. Construct both 98% and 95% confidence intervals for the mean age of the supporters of the candidate for this situation. At each confidence level, comment upon the change in the results of this problem from the results of the previous problem.

            Appendix One: (Age of Supporters)

            40        32        60        58        22        28        66        70        71        55        59        58        62        44        89        48        56        33        46            39        39        44        32        48        49        50        51        18        28        23        34        54        28        76        35        77        38        21            59        51        54        38        45        39        19        90        37        46        22        26        27        39        30        45        27       

4. You also need to estimate the population proportion of supporters of the candidate that are usually loyal supporters of the political party that this candidate represents based upon their attesting to this fact and their previous voting record. What minimum sample sizes will be necessary in order to estimate the desired population proportion under the following conditions?

1. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is thought to equal 80%.
2. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is unknown.
3. The estimate is desired to within ±8% with 98% confidence when the population proportion of supporters of the party is thought to equal 95%.

Comment on the changes in the minimum sample sizes you have computed based upon the changes in the information given in the three parts of this problem.

5. You now need to estimate with 98% confidence the population proportion of supporters of the candidate that describes itself as loyal to the political party represented by the candidate. You randomly sample the population of supporters of the candidate and ascertain whether each one has been a loyal party supporter. The results of that sampling process are shown in appendix two below. Using this information, construct the required confidence interval.

Appendix Two: (Loyal Party Supporter? (Y = yes, N = no))

Y         Y         Y         Y         N         N         Y         Y         Y         Y         N

Y         N         Y         Y         Y         Y         Y         Y         Y         N         Y

N         Y         Y         Y         Y         Y         Y         Y         Y         Y         Y

Y         Y         N         N         N         Y         Y         Y         Y         Y         Y        

Y         Y         Y         Y         Y         Y         N         Y         N         N         Y        

N         Y         Y         Y         Y         Y         Y         Y         Y         N         Y

Y         Y         Y         N         Y         Y         Y         Y         Y         Y         N

N         Y         Y         Y         Y         Y         Y         Y         Y         Y         Y  

3. The weights in pounds of 30 preschool children are listed below. Find the five number summary of the data set.

25 25 26 26.5 27 27 27.5 28 28 28.5

29 29 30 30 30.5 31 31 32 32.5 32.5

33 33 34 34.5 35 35 37 37 38 38

4. A manufacturer receives an order for light bulbs. The order requires that the bulbs have a mean life span of 850hours. The manufacturer selects a random sample of 25 light bulbs and finds they have a mean life span of 845 hours with a standard deviation of 15 hours. Assume the data are normally distributed. Using a 95% confidence level, test to determine if the manufacturer is making acceptable light bulbs and include an explanation of your decision.

5. A manufacturer of golf equipment wishes to estimate the number of left-handed golfers. How large of a sample is needed in order to be 95% confident that the sample proportion will not differ from the true proportion by more than 4%.

6.

A local group claims that the police issue at least 60 speeding tickets a day in their area. To prove their point, they randomly select one month. Their research yields the number of tickets issued for each day. The data are listed below. Assume the population standard deviation is 12.2 tickets. At ? = 0.01, test the group’s claim. Make sure to state your conclusion regarding the claim with your reasoning.

70 48 41 68 69 55 70 57 60 83 32 60 72 58 88 48

59 60 56 65 66 60 68 42 57 59 49 70 75 63 44

7. A local politician, running for reelection, claims that the mean prison time for car thieves is less than the required 4 years. A sample of 80 convicted car thieves was randomly selected, and the mean length of prison time was found to be 3.5 years. Assume the population standard deviation is 1.25 years. At ? = 0.05, test the politician’s claim. Make sure to state your conclusion regarding the claim with your reasoning.

According to a Virginia Tech survey, college students make an average of 11 cell phone calls per day. Moreover, 80% of the students surveyed indicated that their parents pay their cell phone expenses (J. Elliot, “Professor Researches Cell Phone Usage Among Students,” www.physorg.com, February 26, 2007).

1. If you select a student at random, what is the probability that he or she makes more than 10 calls in a day? More than 15? More than 20?

2. If you select a random sample of 10 students, what distribution can you use to model the proportion of students who have parents who pay their cell phone expenses?

3. Using the distribution selected in (c), what is the probability that all 10 have parents who pay their cell phone expenses? At least 9? At least 8?