To investigate the efficacy of a diet, a random sample of 16 male patients is drawn from a population of adult males using the diet. The weight of each individual in the sample is taken at the start of the diet and at a medical follow-up 4 weeks later. Assuming that the population of differences in weight before versus after the diet follows a normal distribution, the t-test for related samples can be used to determine if there was a significant decrease in the mean weight during this period. Suppose the mean decrease in weights over all 16 subjects in the study is 3.0 pounds with the standard deviation of differences computed as 6.0 pounds.

a) The t test should be _______-tailed. (one or two)

b) The computed t statistic is _______.

c) There are _______ degrees of freedom for this test.

d) The critical value for a one-tailed test of the null hypothesis of no difference at the α = 0.05 level of significance is _______.

e) A one-tailed test of the null hypothesis of no difference would _______ (be rejected/not be rejected) at the α = 0.05 level of significance.

The Pat-A-Cake Pastry Shop makes chocolate cake in three sizes – Small, Medium, and Large. For each size, the number of cakes made is an integer (i.e. the shop does not bake only half of a cake). The shop has the following amounts of the three main ingredients on hand – 400 ounces of cake flour, 550 ounces of caster sugar, and 150 ounces of cocoa powder. The table below provides details on the amount of each ingredient required for each cake size as well as the profit contributions. The shop wants to make the appropriate amount of each cake size in order to maximize profit. Cake Small Medium Large Available Plain flour (Ounce) 8 16 21 400 Caster sugar (Ounce) 18 22 25 550 Cocoa powder (Ounce) 3 5 11 150 Profit/Unit $18 $25 $32

a. Develop a spreadsheet model and find the optimal solution using Excel Solver. What is the optimal total profit? Enter your answer without a dollar sign.

b. Based on your answer for a., what quantity of large cakes should be produced to maximize profit contribution? Remember that the number of cakes made should be an integer.

c. Based on your answer for a., what quantity of medium cakes should be produced to maximize profit contribution? Remember that the number of cakes should be an integer.

d. Based on your answer for a., what quantity of small cakes should be produced to maximize profit contribution? Remember that the number of cakes should be an integer.

The amount of syrup that people put on their pancakes is normally distributed with mean 55 mL and standard deviation 11 mL. Suppose that 17 randomly selected people are observed pouring syrup on their pancakes. Round all answers to 4 decimal places where possible.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. If a single randomly selected individual is observed, find the probability that this person consumes is between 53.5 mL and 56.5 mL.
4. For the group of 17 pancake eaters, find the probability that the average amount of syrup is between 53.5 mL and 56.5 mL.
5. For part d), is the assumption that the distribution is normal necessary? YesNo

FIREFLIES (6 pts)

        Your hard work building the fancy fire pit you saw on Pinterest has really paid off! Now every weekend evening in the summer your neighbors and their kids come over to enjoy conversation, games, and cool beverages with you and your family. One especially cunning parent has managed to convince the neighborhood children to put down their tablets and phones and play by catching (and sometimes releasing) fireflies. You make a game out of it by seeing how many they can catch every 15 minutes. Throughout the course of a long evening you collect the data below, which shows how many fireflies the kids catch every 15 minutes. Assuming this data comes from a normally distributed population – can you conclude that the average number of fireflies caught after 10pm was greater than the average caught before 10pm? Use α = 0.05.

Step 1) What type of hypothesis test is required here?                                                                                             

How would you run this test in MINITAB (Menus, Functions used)?                                                           

Is this a left-tailed, right-tailed, or two-tailed test?                                                                                           

Step 2) Verify all assumptions required for this test:                                                                                                       

Step 3) State the null and alternate hypotheses for this test: (use correct symbols and format!)

                   Null hypothesis                                                                                                                                                

                   Alternate hypothesis                                                                                                                                     

Step 4) Run the correct test in MINITAB and provide the information below. Use correct symbols and round answers to 3 decimal places.

Test Statistic                       =                                            Degrees of freedom                       =                           

Critical Value                     =                                            p-value                                                 =                           

Step 5) State your statistical decision (and justify it!)                                                                                                      

                                                                                                                                                                                                                 Step 6) Interpret your decision within the context of the problem: what is your conclusion?

Three experiments investigating the relation between need for cognitive closure and persuasion were performed. Part of the study involved administering a "need for closure scale" to a group of students enrolled in an introductory psychology course. The "need for closure scale" has scores ranging from 101 to 201. For the 83 students in the highest quartile of the distribution, the mean score was x = 177.30. Assume a population standard deviation of σ = 7.69. These students were all classified as high on their need for closure. Assume that the 83 students represent a random sample of all students who are classified as high on their need for closure.

Find a 95% confidence interval for the population mean score μ on the "need for closure scale" for all students with a high need for closure. (Round your answers to two decimal places.)

lower limit'

upper limit

Problem 12. Peter and Paula play a game of chance that consists of several rounds. Each individual round is won, with equal probabilities, by either Peter or Paula; the winner then receives one point. Successive rounds are independent. Each has staked 50 for a total of 100, and they agree that the game ends as soon as one of them has won a total of 5 points; this player then receives the 100. After they have completed four rounds, of which Peter has won three and Paula only one, a fire breaks out so that they cannot continue their game. How should the 100 be divided between Peter and Paula?

Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archaeological site. At one excavation, a sample of 165 potsherds indicated that 78 were of the three-circle, red-on-white pattern.

(a) Find a point estimate p̂ for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern. (Round your answer to four decimal places.)


(b) Compute a 95% confidence interval for the population proportion p of all ceramic potsherds with this distinctive pattern found at the site. (Round your answers to three decimal places.)

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of 5380 permanent dwellings on an entire reservation showed that 1564 were traditional hogans.

(a) Let p be the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answer to three decimal places.)

Give a brief interpretation of the confidence interval.

1% of the confidence intervals created using this method would include the true proportion of traditional hogans.

99% of all confidence intervals would include the true proportion of traditional hogans.   

99% of the confidence intervals created using this method would include the true proportion of traditional hogans.

1% of all confidence intervals would include the true proportion of traditional hogans.

(c) Do you think that np > 5 and nq > 5 are satisfied for this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.    

No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.

No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.

A researcher hypothesizes that short term memory predicts vocabulary knowledge. A random sample of Army recruits is selected and given the digit span (a short term memory test) and vocabulary subtests of the Wechsler Adult Intelligence Scale (WAIS). The data are below. What can be concluded with an α of 0.05?

a) What is the appropriate statistic?
---Select--- na Correlation Slope Chi-Square
Compute the statistic selected above:  

b) Compute the appropriate test statistic(s) to make a decision about H₀.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
p-value =  ; Decision:  ---Select--- Reject H0 Fail to reject H0

c) Using the SPSS results, compute the corresponding effect size(s) and indicate magnitude(s).
If not appropriate, input and/or select "na" below.
Effect size =  ;   ---Select--- na trivial effect small effect medium effect large effect

d) Make an interpretation based on the results.

Better short term memory significantly predicts more vocabulary.Better short term memory significantly predicts less vocabulary.    Short term memory does not significantly predict vocabulary.

Two chemical companies can supply a raw material. The concentration of an element in this material is important. A random sample with 9 observations from company 1 yields a sample mean of 12.6 and a sample standard deviation of 2.7, and another sample with 11 observations from company 2 yields a sample mean of 14.5 and a sample standard deviation of 5.5. Is there a sufficient evidence to conclude that the variance of company 1 is less than variance of company 2 at 5% significance level?

8 percent of all city A residents are known to have abnormal blood pressure requiring some form of medication (give answers to 4 places past decimal)

1. What is the sampling distribution for the sample proportion of citizens requiring medication ina. random sample of 550 people.

Normal Mean:

Standard Deviation:

2. What is the probability that the sample proportion requiring medication, in the random sample of part (1) above, is greater than 10%?

3. What is the minimum sample size necessary for the sample proportion of citizens requiring medication to be approximately normal?

A dog food manufacturer is using a new process to make his dog food.  He believes the new process affects the taste of the food.  To test the new process, the company recruited 9 dogs. The dogs were given the original food and the time required to eat a standard portion of the food was measured. Two days later the same dogs were fed the same quantity of the new formula and the time was measured again. Eating faster is assumed to mean the taste was better.  Eating slower means the taste was worse The times are below. Test the hypothesis that the new process affects the taste of the dog food using an alpha level of .05.

MAKE SURE TO ANSWER ALL OF THE QUESTIONS. SHOW YOUR WORK WHERE POSSIBLE.

a) What is the appropriate test?

b) State the null hypothesis (in words and with means).

c) State the alternative hypothesis (in words and with means).

d) Find the critical value(s) (explain how you found it).

e) Calculate the obtained statistic (show all of your work).

f) Report the results in proper format. Make a decision.

A sample of 220 patients of a family medicine practice was surveyed two weeks after their doctor's visit to ask them whether the symptoms that prompted their visit improved, and whether they complied with the physician's treatment plan. The following table contains the results.

​

Is there a relationship between the two variables?

The marketing director for a cereal company would like to know what proportion of households that receive free samples of cereal with their newspapers later purchase the cereal. A random sample showed 55 out of 100 purchased the cereal after receiving the free sample. Find a 95% confidence interval for the true proportion.

You are interested in constructing a 99% confidence interval for the proportion of all caterpillars that eventually become butterflies. Of the 356 randomly selected caterpillars observed, 42 lived to become butterflies. Round answers to 4 decimal places where possible.

a. With 99% confidence the proportion of all caterpillars that lived to become a butterfly is between  and .

b. If many groups of 356 randomly selected caterpillars were observed, then a different confidence interval would be produced from each group. About  percent of these confidence intervals will contain the true population proportion of caterpillars that become butterflies and about  percent will not contain the true population proportion.