An exercise science major wants to try to use body weight to predict how much someone can bench press. He collects the data shown below on 30 male students. Both quantities are measured in pounds.

a) What type of association does there appear to be between these two variables?
Solved Answer: positive association

b) Compute a 95% confidence interval for the average bench press of 150 pound males. What is the lower limit? Give your answer to two decimal places.

c) Compute a 95% confidence interval for the average bench press of 150 pound males. What is the upper limit? Give your answer to two decimal places.

d) Compute a 95% prediction interval for the bench press of a 150 pound male. What is the lower limit? Give your answer to two decimal places.

e) Compute a 95% prediction interval for the bench press of a 150 pound male. What is the upper limit? Give your answer to two decimal places.

Please provide formulas used for the confidence intervals and prediction intervals and standard error. Thank you!

Although it is generally assumed that routine exercise will improve overall health, there is some question about how much exercise is necessary to produce the benefits. To address this question, a researcher obtained 20 pairs of participants, matched on age, gender, and weight. All participants exercised regularly but one participant within each pair exercised less than two hours per week, and the other spent more than five hours per week exercising. Each participant was subsequently evaluated by a physician and given an overall rating of their health. The researcher runs the statistics using SPSS and obtains the output below:

T-Test

PAIRED SAMPLE STATISTICS

Paired Samples Test

1) using the hypothesis, what is the null hypothesis and the alternative hypothesis? and should it be rejected, why or why not?

2) what is the dependent and independent variable?

3) Is this a between-subjects or within-subjects design?

4) What is the effect size? Please explain your answer.

. Suppose 2% of people have Syndrome X. We have a Syndrome X detecting test which gives which gives a positive result for 90% of people who do have the syndrome, but also gives a positive result for 10% of people who don’t actually have the syndrome. A patient comes in and gets a positive result. What are the chances they have Syndrome X? For full credit, you must show your work.

10. Companies often have different mutual funds to serve different investment time horizons. Use the tab titled BESTFUNDS to determine if there is a difference between in the three-year annualized return for small cap growth, mid-cap growth, and large cap growth mutual funds. (a) Identify which type of test you plan to use and why. (b) Analyze and report your findings. (c) Put the findings into meaningful words (i.e., explain what the test allows you to conclude about the types of mutual funds). (d) Does your result from (b) give you statistical permission to probe group differences, yes or no? (4 points)

9. Mutual funds mix different types of investments which alters performance. Use the tab titled BESTFUNDS1 to determine if there is a difference between the one-year and three-year annualized return for the 20 mutual funds shown in the file. (a) Identify which type of test is most appropriate for you to use, justify your answer. (b) Determine whether or not the mean return differs for the two investment horizons (use α = .05). (c) Make sure to interpret the results for real-world use (i.e., explain what the test allows you to conclude about the mutual funds). (3 points)

A production line manager wants to determine how well the production line is running. He randomly selected 90 items off of the assembly line and found that 8 were defective. (Assume all conditions have been satisfied for building a confidence interval). Find the 99% confidence interval.

Year # AIDS cases diagnosed # AIDS deaths

Graph "year" vs. "# AIDS deaths." Do not include pre-1981. Label both axes with words. Scale both axes. Calculate the following. (Round your answers to the nearest whole number. Round the correlation coefficient r to four decimal places.)

a =

b=

r=

n=

A program was created to randomly choose customers at a clothing store to receive a discount. The program claims 22% of the receipts will get a discount in the long run. The owner of the clothing store is skeptical and believes the program's calculations are incorrect. He selects a random sample and finds that 17% received the discount. The confidence interval is 0.17 ± 0.05 with all conditions for inference met.

Part A: Using the given confidence interval, is it statistically evident that the program is not working? Explain. (3 points)

Part B: Is it statistically evident from the confidence interval that the program creates the discount with a 0.22 probability? Explain. (2 points)

Part C: Another random sample of receipts is taken. This sample is five times the size of the original. Seventeen percent of the receipts in the second sample received the discount. What is the value of margin of error based on the second sample with the same confidence level as the original interval? (2 points)

Part D: Using the margin of error from the second sample in part C, is the program working as planned? Explain. (3 points)

A thermocouple is being calibrated with a well designed ice bath. The measurements are given in the table to the right. Using any mathematical tool of your choice,the mean is .174 and standard deviation of the sample is .24228. I recommend excel for this problem and a simple example is included on the module for this class.

a) What is the bias error in this sample, assuming a perfect ice bath?

b) Assume you have corrected for the bias error and write an expression for the precision error in this sample. Use the formula T=X± ( 2.101)ST (95% ) where X is the sample mean (which will be 0 C after correction for the bias error in this case) ST is the sample standard deviation, and the coefficient 2.101 is a correction to the expression due to the limited amount of data. (Google “t-test” if you are not familiar with this procedure – if we had an infinite amount of data, this value would be 1.96.)   

Test number Measurement C

1 0.220

2 0.280

3 0.190

4 0.510

5    -0.130

6 0.230

7 0.400

8 0.110

9 0.100

10    0.390

11 -0.090

12    0.490

13 0.250

14    0.550

15 0.100

16 -0.180

17 -0.250

18    -0.030

4) In the previous test you were given a formula: The coefficient of the Standard Deviation, 2.101, is based on a test called a t-test and depends on the confidence interval and the number of measurements, which were 95% and 18 in that case. Assume that there are only 14 measurements, but you have exactly the same average and standard deviation that you calculated on the previous problem. Go to section 3.6.12 of the book referenced in problem 1, read that section if necessary, and determine what the coefficient should be for this smaller data set. (In other words, what should “2.101” be replaced with in this new data set with only 14 samples.)

Word Problems with a Sample Data Set:

1. A sample of Full-time SUNY Poly students were asked how many credit hours they were taking this semester and the following information was obtained. Assume credit hours are approximately normally distributed.

16        18        12        17        20        18        15        14

*Round all final answers to 2 decimal places if necessary

1. Construct a 90% confidence interval for the population mean

2. Construct a 95% confidence interval for the population standard deviation

3. SUNY Poly claims that all their students average less than 18 credit hours per semester. Test SUNY’s claim at α = 0.05, what can you conclude about the claim?

1. A company wants to apply k-NN to find out whether new customers response to its marketing campaign or not. We have data of six existing customers and their response. In the table below, we list their response and their distances to two new customers (Jack and Colleen).

What will be the predicted responses of Jack and Colleen using k-NN when k is set to the following values? Justify your answer.

1. k = 1
2. k = 3
3. k = 5

A box in a certain supply room contains four 30W lightbulbs, five 60W lightbulbs, and six 70W lightbulbs. Suppose that four lightbulbs are randomly selected. 1) What is the probability that at least one lightbulb of each type is selected? 2)Find the expected number of lightbulbs (incorrect ones) that need to be selected before a 60W or 70W lightbulb is selected. Show this by building a probability table.

González Industries requires its sales personnel to keep track of their weekly contacts with customers. A sample of 16 reports showed a mean of 32.4 customer contacts per week for the sales personnel, and a sample standard deviation of 5.7 contacts. Assuming customer contacts is a normally distributed variable, generate a 95% confidence interval estimate of the true mean number of customer contacts per week at González Industries. Begin by stating whether this estimation problem should use the student t distribution or the normal (Z) distribution. Should the t distribution be used?

If you indicated the t distribution should be used, give the value that should be used here. If you said, "no", then indicate which Z value should be used.

What is the lower limit of the confidence interval?

What is the upper limit of the confidence interval?

Assume a binomial probability distribution has

p = 0.60

and

n = 300.

(a)

What are the mean and standard deviation? (Round your answers to two decimal places.)

mean standard deviation

(b)

Is this situation one in which binomial probabilities can be approximated by the normal probability distribution? Explain.

No, because np ≥ 5 and n(1 − p) ≥ 5. Yes, because n ≥ 30.     Yes, because np ≥ 5 and n(1 − p) ≥ 5. No, because np < 5 and n(1 − p) < 5. Yes, because np < 5 and n(1 − p) < 5.

(c)

What is the probability of 160 to 170 successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(d)

What is the probability of 190 or more successes? Use the normal approximation of the binomial distribution to answer this question. (Round your answer to four decimal places.)

(e)

What is the advantage of using the normal probability distribution to approximate the binomial probabilities?

The advantage would be that using the normal probability distribution to approximate the binomial probabilities increases the number of calculations. The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations less accurate.     The advantage would be that using the normal probability distribution to approximate the binomial probabilities makes the calculations more accurate. The advantage would be that using the the normal probability distribution to approximate the binomial probabilities reduces the number of calculations.

How would you calculate the probability in part (d) using the binomial distribution. (Use f(x) to denote the binomial probability function.)

P(x ≥ 190) = f(191) + f(192) + f(193) + f(194) +    + f(300)

P(x ≥ 190) = f(190) + f(191) + f(192) + f(193) +    + f(300)

P(x ≥ 190) = f(0) + f(1) +    + f(188) + f(189)

P(x ≥ 190) = f(0) + f(1) +    + f(189) + f(190)

P(x ≥ 190) = 1 − f(189) − f(190) − f(191) − f(192) −    − f(300)

One pharmaceutical company claims their flu vaccine reduces the risk of illness by 40%. One political pundit believes that the real percentage is less. He does a study and the conclusion of the statistical test is to reject the null hypnosis. According to the CDC recent studies show that flu vaccinations reduce the risk of illness by between 40% and 60%. Given this information answer the following questions.

(a) State the null hypothesis as a mathematical statement.

(b) State the alternative hypothesis as a mathematical statement. (c)Was a correct decision made? Why or why not?

(d) If an error was made state what type of error. If no error was made state ”correct decision”.