A population has a mean of 400 and a standard deviation of 50. Suppose a sample of size 125 is selected and x is used to estimate μ.

a. What is the probability that the sample mean will be within +/- 4 of the population mean (to 4 decimals)?

b. What is the probability that the sample mean will be within +/- 11 of the population mean (to 4 decimals)?

A random sample of 11 items is drawn from a population whose standard deviation is unknown. The sample mean is x¯ = 920 and the sample standard deviation is s = 25. Use Appendix D to find the values of Student’s t.

(a) Construct an interval estimate of μ with 95% confidence. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

(b) Construct an interval estimate of μ with 95% confidence, assuming that s = 50. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

(c) Construct an interval estimate of μ with 95% confidence, assuming that s = 100. (Round your answers to 3 decimal places.)
  
The 95% confidence interval is from to

Men’s heights are normally distributed with a mean of 69.5 inches and a standard deviation of 2.4 inches.

1. What percent of men are taller than 5 feet?

2. What percent of men can be a flight attendant if you must be between 5’2” and 6’1”?

3. Find the 45th Percentile of men’s heights.

Let X and Y have the joint PDF

f(x) = { 1/2 0 < x + y < 2, x > 0, y > 0 ;

{ 0 elsewhere

a) sketch the support of X and Y

b) Are X and Y independent? Explain.

c) Find P(x<1 and y<1.5)

Since an instant replay system for tennis was introduced a a major tournament, men challenged 1438 referee calls, with the result that 422 of the calls were overturned. Women challenged 740 referee calls, and 212 of the calls were overturned. Use a .05 significance level to test the claim that men and women have equal success in challenging calls. Complete parts (a) through (c) below. a. test the claim using a hypothesis test. Consider the first sample to be the sample of male tennis players who challenged referee calls and the second sample to be the sample of female tennis players who challenged referee calls. What are the null and alternative hypotheses for the hypothesis test? Identify the test statistic. Z= (round to 2 decimal places as needed) Identify the P-value. P= (round to 3 decimal places as needed) What is the conclusion based on the hypothesis test? The P-value is Less (than/greater) than the significance level of alpha= .05, so (fail to reject/reject) the null hypothesis. There (is sufficient/is not sufficient) evidence to warrant rejection of the claim that women and men have equal success in challenging calls. B. Test the claim by constructing an appropriate confidence interval. The 95% confidence interval is x<(p1-p2)

2. The data sample,below gives the lengths (in cm) of 43 male coyotes found in Nova Scotia. For this x=92.06ands=6.70.

78.0 80.0 90.0 90.5 96.0 96.0,80.0 81.3 83.8 84.5 85.0 86.0 86.4 86.5 87.0 88.0 88.0 88.9 88.9 91.0 91.0 91.0 91.4 92.0 92.5 93.0 93.5 95.0 95.0 95.0 95.0 95.5 96.0 96.0 97.0 98.5 100.0 100.5 101.0 101.6 103.0 104.1 105.0

This data has a bell-shaped distribution. See how well the empirical rule describes the actual distribution of this data by finding the percentage of sample values within one, two, and three standard deviations of the mean.

Q.1 A social psychologist is interested in whether the amount of couple photos (photos featuring the couple together) people post on social media would vary across relationship stages. He administered a questionnaire to a group of couples (one questionnaire for each couple) at the anniversaries of their first year, second year, third year, and fourth year of dating. The questionnaire asked about the total amount of couple photos they posted in social media during the past year. Below are the data, with a higher number denoting more couple photos posted:

a. What are the independent variable and dependent variable of this study?

b. Write down the omnibus null hypothesis and the alternative hypothesis for the overall effect of the independent variable.

c. Conduct a proper statistical test by hand calculation to test the omnibus hypothesis with 5% as the level of significance (α). (For this exercise, the data assumptions of your chosen statistical test can be taken as reasonably met.) Show your calculation formulae and steps. In case you decide to conduct an ANOVA, you are not required to conduct any post-hoc comparisons. Decide whether to reject the null hypothesis or not and state the basis of your decision.

d. Calculate the effect size in terms of eta-squared and omega-squared.

Briefly explain how Levene Test is used. Give an example to illustrate.

Among 16 electrical components exactly 4 are known not to function properly. If 6 components are randomly selected, find the following probabilities: (i) The probability that all selected components function properly. (ii) The probability that exactly 3 are defective. (iii) The probability that at least 1 component is defective.

Listed below are brain volumes (cm3 ) of twins.

Test the claim at the 5% significance level that the brain volume for the first born is different from the second-born twin.

(a) State the null and alternative hypotheses.

(b) Find the critical value and the test statistic.

(c) Should H0 be rejected at the 5% significance level? Make a conclusion.

(d) Construct a 95% confidence interval for the paired difference of the population means

Suppose 70% of all students live in dorms. Of those who live in dorms, 40% of them eat breakfast. Of those who don’t live in dorms, 30% eat breakfast.

5. Make a tree from this information. Label all the probabilities.

6. Make a table from this information. Label everything. Show your work.

7. What percentage of allstudents eat breakfast?

8. What rule/law do we use to answer the previous question?

1. Multiplication rule
2. Law of total probability
3. Bayes rule
4. Addition Rule

9. If a randomly selected student eats breakfast, what is the chance that they live in a dorm?
10. What rule/law do we use to answer the previous question?

1. Multiplication rule
2. Law of total probability
3. Bayes rule
4. Addition Rule

Q.4 A researcher is interested in whether people’s level of loneliness would vary as a function of their relationship status (single vs. in a relationship), and how such difference might depend on whether people own a pet or not. She recruited a group of participants, asking them about their relationship status, pet ownership, and the perceived level of loneliness. The data are as below, with a higher number denoting greater level of loneliness:

a. What are the independent variable(s) and dependent variable of this study?

b. Write down the null hypothesis and the alternative hypothesis for each of the effects in the analysis.

c. Conduct a proper statistical test by hand calculation to test the hypotheses in b., with 5% as the level of significance (α). (For this exercise, the data assumptions of your chosen statistical test can be taken as reasonably met.) Show your calculation formulae and steps. In case you decide to conduct an ANOVA, you are not required to conduct any post-hoc comparisons. Decide whether to reject the null hypothesis or not for each effect and state the basis of your decision

QUESTION 1

1. Assume that the two samples of five cereal boxes (one sample for each of two cereal varieties) listed on the TCCACCTC Web site were collected randomly by organization members. For each sample, assume that the population distribution of individual weight is normally distributed, the average weight is indeed 368 grams and the standard deviation of the process is 15 grams, and obtain the following using Excel and PHStat:

   Weights of Oxford O’s boxes
   360.4
   361.8
   362.3
   364.2
   371.4
   
   Weights of Alpine Frosted Flakes with Vitamins & Minerals boxes
   366.1
   367.2
   365.6
   367.8
   373.5 All answers should be accurate to 2 decimal places.

2. (a) For the Oxford's O:

      (i) The value of the sample mean =

      (ii) The proportion of all samples for each process that would have a sample mean less than the value you calculated in step (a)(i) =

      (iii) The proportion of all the individual boxes of cereal that would have a weight less than the value you calculated in step (a)(i) =

      (iv) The probability that an individual box of cereal will weigh less than 368 grams =

      (v) The probability that 4 out of the 5 boxes sampled will weigh less than 368 grams =

      (vi) The lower limit of the 95% confidence interval for the population average weight =

      (vii) The upper limit of the 95% confidence interval for the population average weight =

   (b) For the Alpine Frosted Flakes:

      (i) The value of the sample mean =

      (ii) The proportion of all samples for each process that would have a sample mean less than the value you calculated in step (b)(i) =

      (iii) The proportion of all the individual boxes of cereal that would have a weight less than the value you calculated in step (b)(i) =

      (iv) The probability that an individual box of cereal will weigh less than 368 grams =

      (v) The probability that 4 out of the 5 boxes sampled will weigh less than 368 grams =

      (vi) The lower limit of the 95% confidence interval for the population average weight =

      (vii) The upper limit of the 95% confidence interval for the population average weight =

QUESTION 2

1. Oxford Cereals then conducted a public experiment in which it claimed it had successfully debunked the statements of groups such as the TriCities Consumers Concerned About Cereal Companies That Cheat (TCCACCTC) that claimed that Oxford Cereals was cheating consumers by packaging cereals at less than labeled weights. Review the Oxford Cereals' press release and supporting documents that describe the experiment at the company's Web site and then answer the following assuming that now you have no information about the mean and standard deviation of the population distribution of the weight of all boxes of the cereal produced:

   Weight
   351.8
   360.65
   372.74
   382.96
   375.28
   352.16
   374.15
   361.8
   366.67
   398.86
   384.34
   367.53
   361.59
   364.47
   382.93
   366.88
   368.14
   408.19
   356.03
   379.27
   380.38
   386.44
   378.72
   342.05
   380.29
   361.1
   355.11
   387
   346.86
   391.94
   366.3
   350.52
   397.27
   349
   373.78
   384.04
   392.55
   361.98
   377.07
   390.88
   395.86
   370.21
   380.66
   389.33
   361.15
   386.74
   353
   354.22
   374.24
   363.77
   352.08
   364.11
   359.79
   367.12
   375.84
   343.29
   357.7
   384.75
   380.72
   356.22
   389.72
   375.28
   380.44
   379.14
   364.64
   379.63
   369.29
   337.1
   371.42
   347.63
   363.86
   381.28
   379.21
   366.26
   365.15
   351.33
   375.91
   363.32
   357.96
   375.58

   All answers should be accurate to 2 decimal places.

   (a) For a two-tailed t-test on whether the population mean weight is equal to 368g:

      (i) The value of the t-test statistic is =

      (ii) The p-value of the t-test statistic is =

      (iii) The lower-critical value is =

      (iv) The upper-critical value is =

   (b) For an upper-tailed t-test on whether the population mean weight is more than 368g:

      (i) The value of the t-test statistic is =

      (ii) The p-value of the t-test statistic is =

      (iii) The upper-critical value is =

   (c) For the 95% confidence interval for the population average weight:

      (i) The lower limit =

      (ii) The upper limit =

Researchers conducted a study to investigate whether there is a difference in mean PEF in children with chronic bronchitis as compared to those without. Data on PEF were collected from 100 children with chronic bronchitis and 100 children without chronic bronchitis. The mean PEF for children with chronic bronchitis was 290 with a standard deviation of 64, while the mean PEF for children without chronic bronchitis was 308 with a standard deviation of 77. Based on the data, is there statistical evidence of a lower mean PEF in children with chronic bronchitis as compared to those without? Run the appropriate test at α=0.05. Assume equal variances. Give each of the following to receive full credit: 1) the appropriate null and alternative hypotheses; 2) the appropriate test; 3) the decision rule; 4) the calculation of the test statistic; and 5) your conclusion including a comparison to alpha or the critical value. You MUST show your work to receive full credit. Partial credit is available.