Suppose that 57% of all college seniors have a job prior to graduation. If a random sample of 80 college seniors is taken, approximate the probability that more than 46 have a job prior to graduation. Use the normal approximation to the binomial with a correction for continuity.

The following data gives the monthly sales (in thousands of dollars) for different advertising expenditures (also in thousands of dollars) and sales commission percentages.

Sales                    245      138      352      322      228      275      560      366

Advertising          16.5     18.0     22.3     18.4     19.0     19.5     30.0     18.6

Commission        10.5     2.0       4.0       3.5       4.5       1.8       9.0       8.5

1. What amount of sales would this model predict for advertising expenditures of 25,000 and sales commission of 8%? [Show your code in “R Code” section. Show the answer in “Answer” section. Leave “Comments” section blank.]
2. Find the correlation coefficients for advertising expenditure and commission compared to sales. Explain the results of these findings. [Show your code in “R Code” section. Show the answer in “Answer” section. Leave “Comments” section blank.]
3. At the 5% level of significance, are advertising expenditure or sales commission percentage significant? Why? [Show your code in “R Code” section. Show the answer in “Answer” section. Include the mathematical notations of two sets of null and alternate hypotheses and the phrase “advertising expenditure” or “sales commission percentage” or “both” along with a justification in a few sentences in “Comments” section.]

A ship is carrying 30 travelers from various great houses on a long sea voyage from Braavos to King’s Landing where they will participate in the Game of Thrones: 5 are from House Targaryen(TAR), and 5 from House Lannister (LAN) 4 are from House Stark (STA) and 4 from House Tyrell (TYR) 3 are from House Baratheon (BAR) and 3 from House Martell (MAR) 1 from each of the following houses: House Arryn (ARR), House Tully (TUL), House Greyjoy (GRE), House Bolton (BOL), House Frey (FRE), House Mormont(MOR) You are now asked to evaluate probabilities relating to this voyage. Part 1 - Leaving the Ship Only 9 of the 30 travelers will end leaving the ship when they reach King’s Landing. Based solely on the number of travelers per house, what is the expected value of the number of travelers from LAN leaving the ship? the number of travelers from STA leaving the ship? the number of travelers from BOL leaving the ship? Part 2 - Game of Thrones The following 9 competitors end up leaving the ship and participating in the Game of Thrones: 3 from TAR, 2 from LAN, 2 from STA, 1 from GRE' and 1 from MAR. At the end of the Game of Thrones,only 3 of the 9 competitors will win titles: 1 will win the Iron Throne (I), 1 will end up being the Hand of the King (H) 1 will be the King of the North (N) Nobody can win more than one title. In how many different ways can these 3 titles be distributed among the 9 competitors? In how many different ways can these 3 titles be distributed among the 5 participating houses? (in other words we are asking you to figure out how many possible combinations of titles by houses there can be.) Your answer should not be derived by listing all the possibilities one by one. Instead you should derive your answer by reasoning with known formulas for permutations and combinations. Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that GRE will win at least one title? Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or their ability to devise a cunning plan, what is the probability that LAN will win at least one title? Based solely on the number of competitors per house and not on their ability to wield a sword, axe, or  ability to devise a cunning plan, what is the probability that TAR will win at least one title?

Stew’s Cars operates 3 dealerships in three regions. The General Manager, Lynn, questioned whether the company’s mean profit margin per vehicle sold differed by region.

Steps

1. Specify population parameter of interest and state the null & alt hypotheses

1. H₀:

H_a:

2. H₀:

H_a:

2. State level of significance & decision rule

3. Select random samples from each population

4. Test hypotheses - Use Excel’s Data Analysis Tool.

Step 5.1            Decision (Reject Ho vs Do not Reject Ho):                       

Step 5.2: Write up the conclusion and implication (use the complete sentence):

Write a research hypothesis for which the independent samples t-test will be appropriate. Clearly identify the independent and dependent variables, and two potential confounding factors.

Low-fat or low-carb? Are low-fat diets or low-carb diets more effective for
weight loss? A sample of 77 subjects went on a low-carbohydrate diet for six
months. At the end of that time, the sample mean weight loss was 4.7 kilograms
with a sample standard deviation of 7.16 kilograms. A second sample of 79
subjects went on a low-fat diet. Their sample mean weight loss was 2.6 kilograms
with a standard deviation of 5.90 kilograms. Can you conclude that the mean
weight loss differs between the two diets? Use α = 0.01.
a) State the hypotheses.
b) Compute the test statistic, t.
c) How many degrees of freedom are there (simple method)?
d) Do you reject H0? Give a concluding statement.

Two types of medication for hives are being tested. The manufacturer claims that the new medication B is more effective than the standard medication A and undertakes a comparison to determine if medication B produces relief for a higher proportion of adult patients within a 30-minute time window. 20 out of a random sample of 200 adults given medication A still had hives 30 minutes after taking the medication. 12 out of another random sample of 200 adults given medication B still had hives 30 minutes after taking the medication. The hypothesis test is to be carried out at a 1% level of significance.

1. State the null and alternative hypotheses in words and in statistical symbols.
2. What statistical test is appropriate to use? Explain the rationale for your answer.
3. Would the test be right-tailed, left-tailed or two-tailed? Explain the rationale for your answer.
4. Describe an outcome that would result in a Type I error. Explain the rationale for your answer.
5. Describe an outcome that would result in a Type II error. Explain the rationale for your answer.

Independent Samples T-test

1. If n1 = 100 and n2 = 100, determine the critical value of t for an independent samples t test, two-tailed alpha = .05.

2. Calculate pooled variance: n1 = 12, s21 = 7.4 and n2 = 13, s22 = 8.2

3. A physician compared the cholesterol levels of a representative sample of Americans who are an American diet vs a representative sample who followed a Mediterranean diet. American diet (control): M = 230, s2 = 24, n = 36; Mediterranean diet (experimental): M = 190, s2 = 26, n = 36. Go through the hypothesis testing steps to write your hypotheses, identify the critical t value, and t value.

4. A nutritionist compared the effectiveness of an online diet program to that of an in-person diet program. After three months, she compared the number of pounds of weight lost. The control (in-person) group lost a mean of 18 pounds (n = 16, s2 = 14.50) and the experimental group (online) lost 16 pounds ( n = 21, s2 = 13.30). Go through the hypothesis testing steps to write your hypotheses, identify the critical t value, and t value.

5 part a) The business college computing center wants to determine the proportion of business students who have personal computers (PC's) at home. If the proportion exceeds 35%, then the lab will scale back a proposed enlargement of its facilities. Suppose 300 business students were randomly sampled and 85 have PC's at home. Find the rejection region for this test using α = 0.10.

A) Reject H0 if z > 1.645 or z < -1.645.

B) Reject H0 if z = 1.28.

C) Reject H0 if z < -1.28.

D) Reject H0 if z > 1.28.

part b) A random sample of 10 parking meters in a resort community showed the following incomes for a day. Assume the incomes are normally distributed. Find the 95% confidence interval for the true mean. $3.60 $4.50 $2.80 $6.30 $2.60 $5.20 $6.75 $4.25 $8.00 $3.00

A) ($1.35, $2.85)

B) ($3.39, $6.01) C)

($2.11, $5.34) D)

($4.81, $6.31)

part c) Health care issues are receiving much attention in both academic and political arenas. A sociologist recently conducted a survey of senior citizens whose net worth is too high to qualify for government health care but who have no private health insurance. The ages of 25 uninsured senior citizens were as follows:

71 76 69 79 89 77 64 92 68 93 72 95 79 65 84 66 71 84 73 76 63 90 78 67 85 Find Q1 of the data.

A) 68

B) 76.5

C) 69

D) 68.5

part d) If one card is drawn from a standard 52 card playing deck, determine the probability of getting a ten, a king or a diamond. Round to the nearest hundredth.

A) 0.37

B) 0.31

C) 0.29

D) 0.40

5. Suppose that researchers study a sample of 50 people and find that 10 are left-handed. (a) Find a 95% confidence interval for the population proportion that is left-handed. (b) What would the confidence interval be if the researchers used the Wilson value ̃p instead? (c) Suppose that an investigator tests the null hypothesis that the population proportion is 18% against the alternative that it is less than that. If α = 0.05 then find the critical value ˆpc. Using ˆp as the sample estimate, would the investigator reject the null? (d) Suppose that researchers are using this critical value but, unbeknownst to them, the true, population proportion is 0.16. Find the power of the test.

a) In order to estimate the population mean, μ, to within 3 at 95% confidence, what is the minimum sample size required? (Assumeσ=6.7).

b) If just the population standard deviation were to increase, then the minimum sample size required would:

decrease /increase   

c)  If just the confidence level were to decrease (i.e. go from 95% to 90% confidence), then the minimum sample size required would:

decrease/increase   

d) If just the bound within which μ was to be estimated were to increase, then the minimum sample size required would:  

decrease/increase

1 part a) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful if the average time spent on the deliveries does not exceed 30 minutes. The owner has randomly selected 17 customers and has delivered pizzas to their homes in order to test if the mean delivery time actually exceeds 30 minutes. Suppose the P-value for the test was found to be 0.0266. State the correct conclusion.

A) At α = 0.05, we fail to reject H0.

B) At α = 0.03, we fail to reject H0.

C) At α = 0.02, we reject H0.

D) At α = 0.025, we fail to reject H0.

part b) According to insurance records a car with a certain protection system will be recovered 93% of the time. Find the probability that 5 of 9 stolen cars will be recovered.

A) 0.93

B) 0.002

C) 0.07

D) 0.556

part c) Numbered disks are placed in a box and one disk is selected at random. If there are 6 red disks numbered 1 through 6, and 4 yellow disks numbered 7 through 10, find the probability of selecting a yellow disk, given that the number selected is less than or equal to 3 or greater than or equal to 8.

A) 3/5

B) 1/2

C) 3/4

D) 3/10

part d) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 5.0 minutes and a standard deviation of 1 minute. Find the cut-off time which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot.

A) 5.5 min

B) 5.8 min

C) 5.7 min

D) 5.3 min

1. Over a sample of 47 voyages. Mr. Sulu found the Starship Enterprise traveled a mean of 658 light-years-per-dilithium-crystal (lypdc) with a standard deviation of 76 lypdc.

(a) Find a point estimate for the mean lypdc on all starship voyages.



(b) Find the 95% margin of error for the estimate in (a).






(c) Make a 95% confidence interval for the mean lypdc on all starship voyages. Interpret the interval.

The design of an automotive gear requires the diameter to be 4 inches. To evaluate the production quality, you sample 30 gears and find the average to be 4.11 with a standard deviation of 0.67. Construct a hypothesis test at a 0.05 level of significance to determine if the gear diameter is larger than 4 inches.

Select one:

a. The critical value is 2.05, the test statistic is 0.90, conclude the gear diameter is not equal to 4 inches

b. The critical value is 2.05, the test statistic is 0.90, conclude the gear diameter is greater than 4 inches

c. The critical value is 2.05, the test statistic is 0.90, conclude the gear diameter is less than or equal to 4 inches

d. The critical value is 1.70, the test statistic is 0.90, conclude the gear diameter is less than or equal to 4 inches

e. The critical value is 1.70, the test statistic is 0.90, conclude the gear diameter is greater than 4 inches

Let x be a random variable representing percentage change in neighborhood population in the past few years, and let y be a random variable representing crime rate (crimes per 1000 population). A random sample of six Denver neighborhoods gave the following information.

In this setting we have Σx = 70, Σy = 586, Σx² = 1136, Σy² = 71,796, and Σxy = 8844.

(a) Find x, y, b, and the equation of the least-squares line. (Round your answers for x and y to two decimal places. Round your least-squares estimates to four decimal places.)

(b) Draw a scatter diagram displaying the data. Graph the least-squares line on your scatter diagram. Be sure to plot the point (x, y).

(c) Find the sample correlation coefficient r and the coefficient of determination. (Round your answers to three decimal places.)

What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.)
%

(d) Test the claim that the population correlation coefficient ρ is not zero at the 10% level of significance. (Round your test statistic to three decimal places and your P-value to four decimal places.)

Conclusion

Reject the null hypothesis, there is sufficient evidence that ρ differs from 0.Reject the null hypothesis, there is insufficient evidence that ρ differs from 0.    Fail to reject the null hypothesis, there is sufficient evidence that ρ differs from 0.Fail to reject the null hypothesis, there is insufficient evidence that ρ differs from 0.

(e) For a neighborhood with x = 19% change in population in the past few years, predict the change in the crime rate (per 1000 residents). (Round your answer to one decimal place.)
crimes per 1000 residents

(f) Find S_e. (Round your answer to three decimal places.)
S_e =

(g) Find a 95% confidence interval for the change in crime rate when the percentage change in population is x = 19%. (Round your answers to one decimal place.)

(h) Test the claim that the slope β of the population least-squares line is not zero at the 10% level of significance. (Round your test statistic to three decimal places and your P-value to four decimal places.)

Conclusion

Reject the null hypothesis, there is sufficient evidence that β differs from 0.Reject the null hypothesis, there is insufficient evidence that β differs from 0.    Fail to reject the null hypothesis, there is sufficient evidence that β differs from 0.Fail to reject the null hypothesis, there is insufficient evidence that β differs from 0.

(i) Find a 95% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)

Interpretation

For every percentage point increase in population, the crime rate per 1,000 increases by an amount that falls within the confidence interval.

For every percentage point increase in population, the crime rate per 1,000 increases by an amount that falls outside the confidence interval.  

  For every percentage point decrease in population, the crime rate per 1,000 increases by an amount that falls within the confidence interval.

For every percentage point decrease in population, the crime rate per 1,000 increases by an amount that falls outside the confidence interval.