Waiting time for checkout line at two stores of a supermarket chain were measured for a random sample of customers at each store. The chain wants to use this data to test the research (alternative) hypothesis that the mean waiting time for checkout at Store 1 is lower than that of Store 2.
Store 1 (in Seconds)
Store 2 (in Seconds)

461
264

384
308

167
266

293
224

187
244

115
178

195
279

280
289

228
253

315
223

205

197

What are the null and alternative hypothesis for this research?

What are the sample mean waiting time for checkout line and the sample standard deviation for the two stores?

Compute the test statistic t used to test the hypothesis. Note that the population standard deviations are not known and therefore you cannot use the formula in Section 10.1. Use the one in Section 10.2 instead.

Compute the degree of freedom for the test statistic t.

Can the chain conclude that the mean waiting time for checkout at Store 1 is lower than that of Store 2? Use the critical-value approach and α = 0.05 to conduct the hypothesis test.

Construct a 95% confidence interval for the difference of mean waiting time for checkout line at the two stores.

To compare prices of two grocery stores in Toronto, a random sample of items that are sold in both stores were selected and their price noted in the first weekend of July 2018:

Item
Store A
Store B
Difference (Store A - Store B)

1
1.65
1.99
-0.34

2
8.70
8.49
0.21

3
0.75
0.90
-0.15

4
1.05
0.99
0.06

5
11.30
11.99
-0.69

6
7.70
7.99
-0.29

7
6.55
6.99
-0.44

8
3.70
3.59
0.11

9
8.60
8.99
-0.39

10
3.90
4.29
-0.39

What are the null and alternative hypothesis if we want to confirm that on average, prices at Store 1 is different from the prices at Store 2, that is, the difference is different from 0?

What are the sample mean difference in prices and the sample standard deviation?

Compute the test statistic t used to test the hypothesis.

Compute the degree of freedom for the test statistic t

Can we conclude that on average, prices at Store 1 is different from the prices at Store 2? Use the critical-value approach and α = 0.05 to conduct the hypothesis test.

Use the above data to construct a 95% confidence interval for the difference in prices between the two stores.

3. In a completely randomized design, 7 experimental units were used for each of the three levels of the factor. (Total: 6 marks; 2 marks each)

Source of Variation
Sum of Squares
Degrees of Freedom
Mean Square
F

Treatment

Error
432076.5

Total
675643.3

Complete the ANOVA table.

Find the critical value at the 0.05 level of significance from the F table for testing whether the population means for the three levels of the factors are different.

Use the critical value approach and α = 0.05 to test whether the population means for the three levels of the factors are the same.

22 randomly selected students were asked the number of movies they watched the previous week.

The results are as follows: # of Movies 0 1 2 3 4 5 6 Frequency 4 1 1 5 6 3 2

Round all your answers to 4 decimal places where possible.

The mean is:

The median is:

The sample standard deviation is:

The first quartile is:

The third quartile is:

What percent of the respondents watched at least 2 movies the previous week? %

78% of all respondents watched fewer than how many movies the previous week?

For the following description of​ data, identify the​ W's, name the​ variables, specify for each variable whether its use indicates it should be treated as categorical or quantitative and for any quantitative variable identify the units in which it was measured. Determine if the data comes from a designed survey or experiment. Determine if the variables are time series or​ cross-sectional.

A company surveyed a random sample of

65006500

employees in the region. One question they asked​ was, "If your employer provides you with mentoring opportunities are you likely to remain in your job for the next

tenten

​years?" They found that

580580

members of the sample said yes.

Identify the Who for this study.

A.The

65006500

employees in the region

B.The

580580

employees who answered yes

C.

All the employees in the region

D.

All people in the region

Identify the What for this study. Select all that apply.

A.

The amount an employee receives in benefits

B.

The average amount of time an employee remains at their job

C.Whether or not an employee is likely to remain at their job for the next

tenten

years

D.The average number of jobs someone goes through in a

fivefive

year period

E.The average number of jobs someone goes through in a

tenten

year period

F.Whether or not an employee is likely to remain at their job for the next

fivefive

years

Identify the When for this study.

A.Over the course of

fivefive

years

B.Within the past

tenten

years

C.

Over the course of a year

D.

This information is not given.

Identify the Where for this study.

A.

Over the phone

B.

At a hotel

C.

Online

D.

This information is not given.

Identify the Why for this study.

A.To determine the least amount of benefits to keep an employee at their job for a minimum of

tenten

years.

B.To determine how many people have been at their current job for at least

fivefive

years.

C.To determine the likelihood that someone remains at their job for the next

tenten

years given that their employer provides them with mentoring opportunities.

D.

This information is not given.

Identify the How for this study.

A.

Employers went door to door surveying residents.

B.

The company passed out a survey to be filled out during the work day.

C.

An online poll was posted on the​ company's website.

D.

This information is not given.

Specify the categorical variables for this study. Select all that apply.

A.

Amount of time at the company

B.

Gender

C.

Age

D.

Mentoring opportunities

E.Whether or not an employee is likely to remain at their job for the next

tenten

years

F.

There are no categorical variables.

Specify the quantitative variables and their units for this study. Select all that apply.

A.

Number of​ jobs; count

B.

Amount of time at the​ company; years

C.

​Age; years.

D.

Mentoring​ opportunities; count

E.

There are no quantitative variables.

Specify whether the data come from a designed survey or experiment.

Experiment

Designed survey

Are the variables time series or​ cross-sectional?

​Cross-sectional

Time series

For the following description of​ data, identify the​ W's, name the​ variables, specify for each variable whether its use indicates it should be treated as categorical or​ quantitative, and for any quantitative variable identify the units in which it was measured​ (or note that they were not​ provided). Specify whether the data come from a designed survey or experiment. Are the variables time series or​ cross-sectional? Report any concerns you have as well.

A certain horse race has been run every year since

18711871

in a city. The accompanying table shows the official race data for the first two races and two recent races.

Identify the​ "who." Choose the correct answer below.

A.

The horses that competed in the​ city's horse races

B.

The​ city's horse races

C.

Horse races

D.

This information is not given.

Identify the​ "what." Choose the correct answer below.

A.

​Year, winner,​ margin, jockey,​ winner's payoff,​ duration, track condition

B.

​Year, margin,​ winner's payoff, duration

C.

​Winner, jockey, track condition

D.

This information is not given.

Identify the​ "when." Choose the correct answer below.

A.

May

B.

18711871​-20052005

C.

18711871​,

18721872​,

20042004​,

and 20052005

D.

This information is not given.

Identify the​ "where." Choose the correct answer below.

A.

All race tracks in the state

B.

The city where the horse races took place

C.

United States

D.

This information is not given.

Identify the​ "why." Choose the correct answer below.

A.

To see if the same horse won in multiple years

B.

To maintain a list of the winners

C.

To compare the times of the winners from year to year

D.

This information is not given.

Identify the​ "how." Choose the correct answer below.

A.

A chronic better recorded the data.

B.

A random sample of races was taken.

C.

Official statistics were collected at the time of the race.

D.

This information is not given.

Specify the categorical variables for this problem. Select all that apply.

A.

Winner

B.

Jockey

C.

Track Condition

D.

​Winner's payoff

E.

Duration

F.

Year

G.

Margin

H.

There are no categorical variables.

Specify the quantitative variables and identify the units for this problem. Select all that apply.

A.

​Year; the units are years

B.

​Winner's payoff; the units are dollars

C.

​Winner; the units not specified

D.

Track​ condition; the units not specified

E.

​Jockey; the units not specified

F.

​Duration; the units are minutes and seconds

G.

​Margin; the units are lenghts

H.

There are no quantitative variables.

Specify whether the data come from a designed survey or experiment. Choose the correct answer below.

A.

Experiment

B.

Designed survey

C.

This information cannot be determined by the given data.

Are the variables time series or​ cross-sectional?

Time series

​Cross-sectional

Neither

Specify any concerns. Select all that apply.

A.

The data sample size was too large.

B.

There are too many quantitative variables.

C.

The data collection was done incorrectly.

D.

There are no specific concerns.

Given the following numbers: 25 16 61 18 15 20 15 20 24 17 19 28, derive the mean, median, mode, variance, standard deviation, skewness, kurtosis, range, minimum, maximum, sum, and count. Interpret your results. What is the empirical rule for two standard deviations of the data?

The lowest and highest observations in a population are 14 and 48, respectively. What is the minimum sample size n required to estimate μ with 90% confidence if the desired margin of error is E = 1.5? What happens to n if you decide to estimate μ with 95% confidence? (You may find it useful to reference the z table. Round intermediate calculations to at least 4 decimal places and "z" value to 3 decimal places. Round up your answers to the nearest whole number.)

John is lying on the sidewalk after robbing a bank, in pain and mulling over how to quantify the uncertainty of his survival, when Dirty Harry walks over. Dirty Harry pulls out his 44 Magnum and puts two bullets opposite each other in the six slots in the cylinder (e.g., if you number them 1 .. 6 clockwise, he puts them in 1 and 4), spins the cylinder randomly, and, saying "The question is, are you feeling lucky, probabalistically speaking, computer science punk?" points it at John head and pulls the trigger.... "CLICK!" goes the gun (no bullet) and Dirty Harry smiles... "How about that .... Let's see if this gun is memory-less!" Without spinning the cylinder again, he points the gun at Wayne's head and pulls the trigger again.

(a) What is the probability that (at least in my dream) John is hit?

(b) Now, suppose that when Dirty Harry put the bullets in the gun, he put them right next to each other (e.g., in slots 1 and 2). He spins it as usual. What is the probability in this case John is hit?

(c) Suppose Dirty Harry puts the bullets in two random positions in the cylinder and we don't have any idea where they are. He spins it as usual. Now what is the probability that John will be hit?

The firm that manufactures patriot missiles purchases the guidance circuits from three different suppliers. Supplier A provides 30% of the guidance circuits and those circuits have a fault probability of pA =0.02, whereas the circuits from supplier B, which provides 25% of those purchased, have a fault probability of pB =0.025. The guidance circuits purchased from supplier C have pC =0.01. If a batch of 200 missiles is fired during a particular strategic offensive and three of the missiles fail to track to target, what is the probability that the batch of missiles contained guidance circuits obtained from supplier B?

Problem 6 (Inference via Bayes’ Rule)
Suppose we are given a coin with an unknown head probability θ ∈ {0.3,0.5,0.7}. In order to infer the value θ, we experiment with the coin and consider Bayesian inference as follows: Define events A1 = {θ = 0.3}, A2 = {θ = 0.5}, A3 = {θ = 0.7}. Since initially we have no further information about θ, we simply consider the prior probability assignment to be P(A1) = P(A2) = P(A3) = 1/3.
(a) Suppose we toss the coin once and observe a head (for ease of notation, we define the event B = {the first toss is a head}). What is the posterior probability P(A1|B)? How about P(A2|B) and P(A3|B)? (Hint: use the Bayes’ rule)
(b) Suppose we toss the coin for 10 times and observe HHTHHHTHHH (for ease of notation, we define the event C = {HHTHHHTHHH}). Moreover, all the tosses are known to be independent. What is the posterior probability P(A1|C), P(A2|C), and P(A3|C)? Given the experimental results, what is the most probable value for θ?
(c) Given the same setting as (b), suppose we instead choose to use a different prior probability assignment P(A1) = 2/5,P(A2) = 2/5,P(A3) = 1/5. What is the posterior probabilities P(A1|C), P(A2|C), and P(A3|C)? Given the experimental results, what is the most probable value for θ?

1. Stacy and Leslie are playing a very simple gambling game. They toss a coin and Stacy wins if it comes up “heads” while Leslie wins if it comes up “tails.” After 12 hours of gambling, Leslie begins to suspect that Stacy has been cheating because Stacy has won more games. Leslie accuses Stacy, but Stacy pleads innocent and proposes to test Leslie’s claim by doing an experiment in which the coin is tossed 14 times.

1. State the null and alternative hypotheses for this experiment.
2. If α = .01, what is the rejection region?
3. They do the experiment and “heads” occurs 8 times. What can they conclude? What if heads comes up 12 times?
4. Let’s assume that Stacy actually is cheating because she is using a coin that is biased to come up “heads” 55% of the time. What was the power of the experiment they did?
5. Given the power of the experiment, was Stacy clever or stupid to propose it as a test to Stacy?

6. A school system has a high rate of turn-over among new teachers. Specifically, 30% of the teachers that are hired leave within 2 years. The superintendent is concerned about the problem and institutes a program of teacher mentoring that he hopes will improve retention of the teachers. After the first 2 years of the program, he evaluates whether it is working by recording what happened with the 16 teachers who were hired at the start of the program. He finds that 3 of original 16 have left.

a. Complete the relevant hypothesis test, using α = .05.

b. Suppose that the mentoring program actually does improve retention to the point where the true probability of a teacher leaving is actually 10%. What was the power of the principal’s study? What does the number you compute mean in English? Explain the relevance (or lack of relevance) of your power calculation to your conclusion in part ‘a.’

1. The production manager of a company that produces an over-the-counter cold remedy wants to boost sales of the product. The product is considered effective by the people who have tried it, but many people decide not to buy it again because it tastes like day-old soapsuds. The manufacturer is trying to decide whether to add a lemon flavor to the product. Because the flavoring will increase production costs, the manager wants to be certain that people respond favorably to the flavoring before using it. Twenty people with colds are randomly sampled. On two difference occasions, each person uses the product and indicates which version (taste) is preferred. (They must pick one.)

1. State the competing hypotheses and the rejection region for α = .01.
2. What conclusion should be made if 16 people prefer the lemon-flavored product? What if 2 people prefer the lemon flavor?
3. Why would using α = .01 make sense in this case? If the results indicate that the null hypothesis should be rejected in favor of the alternative hypothesis, what additional problem(s) of interpretation might the production manager have to face?

Students will follow the hypothesis testing steps for each problem. They will compute the problem using the SPSS program. They will write the results in appropriate APA format and interpret the results. Steps of hypothesis testing will be typed out in a word document, as well as a copy and paste of the SPSS output.

For the following problems, you will:

• Do all steps of hypothesis testing
  • Populations and hypotheses
  • Write out the steps you would do to calculate t
  • Choose the t-cutoff score
  • Calculate the t-statistic using the computer program SPSS
  • Write the t-statistic using proper APA format
  • Decide whether or not you would reject the null hypothesis
• Interpret this result
• Be sure to include a copy of the SPSS output in the word document

1. Single Sample T-test

A researcher would like to study the effect of alcohol on reaction time. It is known that under regular circumstances the distribution of reaction times is normal with μ = 200. A sample of 10 subjects is obtained. Reaction time is measured for each individual after consumption of alcohol. Their reaction times were: 219, 221, 222, 222, 227, 228, 223, 230, 228, and 232. Use α = 0.05.

A social media survey found that 71%

of parents are​ "friends" with their children on a certain online networking site. A random sample of 140

parents was selected. Complete parts a through d below.

a. Calculate the standard error of the proportion.

sigma Subscript p

equals0.0383

​(Round to four decimal places as​ needed.)

b. What is the probability that

105

or more parents from this sample are​ "friends" with their children on this online networking​ site?

​P(105

or more parents from this sample are​ "friends" with their

​children)equals

nothing

​(Round to four decimal places as​ needed.)

1. According to the empirical rule, for a distribution that is symmetric and bell-shaped, approximately _______ of the data values will lie within 3 standard deviations on each side of the mean.

2. Assuming that the heights of boys in a high-school basket- ball tournament are normally distributed with mean 70 inches and standard deviation 2.5 inches, how many boys in a group of 40 are expected to be taller than 75 inches?

3. Let x be a random variable that represents the length of time it takes a student to complete Dr. Gill’s chemistry lab project. From long experience, it is known that x has a normal distribution with mean μ = 3.6 hours and standard deviation σ = 0.5.

Convert each of the following x intervals to standard z intervals.

(a) x ≥ 4.5 4. (a) __________________________

(b) 3 ≤ x ≤ 4 (b) __________________________

(c) x ≤ 2.5 (c) __________________________

Convert each of the following z intervals to raw-score x intervals.

(d) z ≤ −1 (d) __________________________

(e) 1 ≤ z ≤ 2 (e) __________________________

(f) z ≥ 1.5 (f)_ __________________________