Stats 250 W16 HW 6 complete solutions correct answers key

Stats 250 W16 HW 6 complete solutions correct answers key

Question 1 Background :

Relationship satisfaction

We are interested in studying the relationship satisfaction of young adults before and after they go off to college

and are separated from their significant other. A random sample of 35 young adults who are about to be

separated from their significant other by going to college was collected. Each participant was asked a series of

questions, which was then converted to a satisfaction score ranging from 0-50 points (a higher score indicates

greater satisfaction). The satisfaction score for each participant was measured before leaving for college and

after spending the first semester at college away from their significant other. For each participant, the difference

in satisfaction score was computed (satisfaction score after minus satisfaction score before). The 90%

confidence interval for the population mean difference in satisfaction score for all such young adults who

are separated from their significant other after one semester of college is given by: -9.0 points to -0.9 points.

Question 1 Subquestions

1.a

2 point(s)

Based on the provided 90% confidence interval, do the data support that the population mean satisfaction

score significantly changed after going off to college for the population of all such young adults (at the 10%

significance level)? State your answer of Yes or No and provide a brief explanation in context.

Student answer

No answer entered.

1.b

0 point(s)

Is the following statement True or False? About 90% of all such young adults who separate from their significant

other to attend college will have a decrease in satisfaction score between 0.9 and 9.0 points.

True

False

1.c

0.5 point(s)

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Is the following statement True or False? If this study were repeated many times, with random samples of 35

young adults who separate from their significant other, and for each sample a corresponding 90% confidence

interval was made, 90% of the resulting intervals are expected to contain the sample mean difference in

satisfaction score.

True

False

1.d

0 point(s)

Is the following statement True or False? The Central Limit Theorem states that since the sample is large

enough (larger than 25 or 30), then we can conclude that the difference in satisfaction scores (difference = After -

Before) for the population of all such young adults follows approximately a normal distribution.

True

False

Question 2 : 1 point(s)

Confidence Level for population mean difference in time

A professor wanted to know if the amount of time his students spent on Facebook exceeded the time they spent

studying, on average. From one of his large lectures of 200 students, he took a random sample of 20 students

and asked them to report the number of hours they spent on Facebook in a week and the number of hours they

spent studying in that same week. He computes a 90% confidence interval for the population mean difference

(difference computed as Facebook time - studying time). His resulting confidence interval is (-5.2, -0.3). The

90% confidence level is: (select all that are correct)

Question 2 Multiple Choice Options

the probability the procedure will provide a 90% confidence interval that covers the sample mean difference

in time (Facebook time - studying time), if the study were repeated in the future.

the probability of making a Type 1 error if the 90% confidence interval is used to test the null hypothesis

that the population mean difference in time (Facebook time - studying time) is zero.

the probability that students in the population will have a difference in time (Facebook time - studying time)

that falls in the 90% confidence interval.

the probability the procedure will provide a 90% confidence interval that covers the population mean

difference in time (Facebook time - studying time), if the study were repeated in the future.

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none of the above are correct

Question 3 Background :

Measuring E. coli Levels

Agricultural researchers wish to compare two methods for measuring the level of E. coli bacteria contamination in

beef. Two different methods (A and B) were used on each of ten randomly selected specimens of a certain type

of beef. The resulting measurements (in milimicrobes/liter) can be found in the data set EColi.RData (https://pbjcoursework.

s3.amazonaws.com/Intro%20to%20Statistics/instructor/3-6-2016/1_55_267-EColi.RData).

Question 3 Subquestions

3.a

2 point(s)

You have been asked to analyze the data. In particular, you are asked to estimate the population mean

difference in E. coli level. Use R (R Commander) to compute a new variable difference = MethodB – MethodA.

Then use R to compute those summary statistics for the differences. Be sure to include the mean, standard

deviation, IQR, five number summary, and the standard error of the mean. Copy and paste in those descriptive

statistics from the R output in your text answer box. Below the R output, provide the sample mean difference in

E. coli level and the standard error of the mean, including units.

Student answer

No answer entered.

3.b

0 point(s)

Use R (R commander) to provide a 95% confidence interval estimate for the population mean difference in E.

coli measurements (where difference = Method B - Method A). Copy and paste in R output in your text answer

box. Below the R output, provide the 95% confidence interval including units.

Student answer

No answer entered.

3.c

0.5 point(s)

Based on the confidence interval, do the two methods appear to be significantly different (at the 5% level) in

terms of measuring E. coli, on average?

Yes, since the interval is quite narrower and does not contain very many reasonable values.

No, since 95% confidence interval does contain the value of zero.

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Question 4 : 1 point(s)

What does it mean?

You interview 100 University of Michigan students in their freshman year and again in their senior year and ask

each about the typical number of minutes per day that they spend using social media. Let μ represent the

population mean difference in time spent using social media (freshman - senior). You wish to test the hypotheses

H : μ = 0 versus H : μ ≠ 0. You compute the 95% confidence interval for μ and find it to be (13.2 minutes,

24.8 minutes). Assume the necessary conditions for conducting the confidence interval and test are met. Which

of the following are correct statements? Select all that are correct.

Question 4 Multiple Choice Options

d

0 d a d d

You would reject the null hypothesis at the 5% level.

Your would fail to reject the null hypothesis at the 5% level.

The p-value for the test is more than 0.05.

A mistake has been made as the confidence interval should contain 0 about 95% of the time.

Question 5 : 1 point(s)

Statistical Significance

Which of the following statements about statistical significance is (are) correct? Select all that are correct.

Question 5 Multiple Choice Options

If a result is statistically significant at the 5% level, it is also significant at the 1% level.

If a result is statistically significant at the 5% level, it is also significant at the 10% level.

If we select a 5% level, we are requiring that the data provide evidence against the null hypothesis so strong

that seeing data that strong or stronger would happen no more than 5% of the time when the null

hypothesis is true.

If the null hypothesis is true, then the significance level is also the probability of rejecting the null hypothesis.

The significance level is the probability that the null hypothesis is true.

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Question 6 Background :

Was it Colder?

A meteorologist wants to determine if it was colder this past October than it was in October of the previous year,

on average, using a 10% significance level. From each October month, he randomly selects 8 different days

from each year and records the temperature. Let Group 1 = temperatures from October this past year and

Group 2 = temperatures from October the previous year.

Question 6 Subquestions

6.a

2 point(s)

Which of the following is the appropriate symbol to complete set of hypotheses?

H : μ 0 1 = μ2 versus Ha: μ1 ? μ2

> (greater than or one-sided to the right)

< (less than or one-sided to the left)

not equal to (two-sided)

6.b

0 point(s)

Suppose the meteorologist conducts the appropriate test and obtains a p-value of 0.076 and thus decides to

reject the null hypothesis. What type of error could he have made?

Type 1 error

Type 2 error

Since the decision has been made, no error is possible.

Question 7 : 1 point(s)

Paired t-Test Assumptions

To use a paired t confidence interval to estimate the population mean difference in responses, or to perform a

paired t test regarding hypotheses about the population mean difference, there are some assumptions (or

conditions). Which of the following is a (are) required assumption(s)? Select all that apply.

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Question 7 Multiple Choice Options

The two sets of measurements that are used to form the pairs are independent random samples.

The number of paired observations (or differences) is more than 30.

The distribution of the population of all differences in responses is normal.

The standard deviation for the population of all differences is known.

Question 8 Background :

Cell Phones and Driving

An experiment was designed to assess whether the use of cell phones during driving generally increases a

driver’s reaction time. A random sample of 32 students was selected for the study. Each student was tested

twice for reaction time in a simulated driving experience, once without an ongoing cell phone conversation and

once with an ongoing cell phone conversation. Half of the students were randomly selected to have a reaction

time assessment within the first three minutes without a cell phone conversation followed by a reaction time

assessment again in the next three minutes with a cell phone conversation. For the other half, the order of the

two conditions (without versus with) was reversed.

Many factors were carefully controlled for during the experiment. For example, the instructions given to each

student were the same, during the with cell phone assessment the length of conversation with the caller and the

script for the conversation was the same. The reaction time was measured as the time between a traffic light

turning red and the student engaging the brake (to the nearest millisecond). The resulting data are given in the

data set named CellPhone.Rdata (https://pbj-coursework.s3.amazonaws.com/files/11-3-2015/4_17_574-

CellPhone.RData).

Question 8 Subquestions

8.a

2 point(s)

The researchers wished to assess if the reaction time would increase with the use of a cell phone on average for

the population of all student drivers. Let the parameter μ represent the population mean difference in reaction

time (without cell less with cell) for the population of all student drivers. The researcher knows that the null

hypothesis of no difference on average should be expressed as H : μ = 0. Which direction is the appropriate

one to correctly complete the alternative hypothesis H : μ __ 0?

d

0 d

a d

greater than (>)

less than (<)

not equal to

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8.b

0 point(s)

Generate the R paired samples t test output. Copy and paste the test results from R Commander in for your

answer.

Below the results state the test statistic value and the corresponding p-value.

Test statistic t = _________ and p-value = __________

Student answer

No answer entered.

8.c

0.5 point(s)

Here are some additional summary measures of the differences in reaction times.

> numSummary(CellPhone[,"difference"], statistics=c("mean", "sd", "se(mean)",

+ "IQR", "quantiles"), quantiles=c(0,.25,.5,.75,1))

mean sd se(mean) IQR 0% 25% 50% 75% 100% n

-50.625 52.48579 9.278264 75.25 -150 -84.75 35.5 -9.5 24 32

Which of the following (slightly rounded) values from this summary output correctly completes this interpretation

statement?

We would estimate the possible sample mean differences in reaction time ( values) to be about ______

milliseconds away from the population mean difference in reaction times μd, on average.

-50.6

52.5

9.3

8.d

0 point(s)

What distribution is used to find the p-value for the test?

standard normal N(0,1) distribution

a t(31) distribution

a t(32) distribution

8.e

0 point(s)

Based on the test results, what decision and conclusion should be made at a 10% significance level? Select one:

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Reject the null hypothesis and conclude there is insufficient evidence to demonstrate that the use of a cell

phone increases reaction time on average for the population of student drivers represented by this sample.

Reject the null hypothesis and conclude there is sufficient evidence to demonstrate that the use of a cell

phone increases reaction time on average for the population of student drivers represented by this sample.

Fail to reject the null hypothesis and conclude there is insufficient evidence to demonstrate that the use of a

cell phone increases reaction time on average for the population of student drivers represented by this

sample.

Fail to reject the null hypothesis and conclude there is sufficient evidence to demonstrate that the use of a

cell phone increases reaction time on average for the population of student drivers represented by this

sample.

8.f

0 point(s)

Someone on the research team wondered if the study could have been conducted as two independent samples

(with 32 subjects per condition). The director of the study responded such a study was possible, but that by

using each patient as his own control in their study, they had reduced the variability of the response (and, hence,

reduced the necessary sample size).

The research team member also remembered that one assumption for this test to be valid is that the differences

in reaction times can be considered a random sample from the population of interest. The provided graph was

made for checking another required assumption to conduct the hypothesis test.

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Clearly state the assumption that this graph helps to assess (in context).

Student answer

No answer entered.

Question 9 Background :

Inter-Rater Reliability

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Two different technicians (Tech A and Tech B) each measured the cardiac output for a sample of 36 patients

using Doppler Echocardiaography. The supervisor would like to use this paired data to assess inter-rater

reliability by testing if the population mean difference in cardiac output is 0 (no difference on average) versus the

alternative that there is a difference. The hypotheses to be tested are: H : μ = 0 versus H : μ ≠ 0, using a 10%

significance level.

Question 9 Subquestions

0 d a d

9.a

2 point(s)

The sample mean difference in cardiac output (Tech A less Tech B) was 2.39 standard errors below 0. Provide a

complete sketch of the p-value that the supervisor can include in her report. You can use the pval() function in R

which will give the exact p-value (and allow you to add your name to the graph when you type 'y' to save the

graph to your desktop), or make a complete by hand sketch and provide the bounds for the p-value. Make sure

to include a title for and include your name with your graph.

Student answer

No answer entered.

9.b

0 point(s)

You remember some condition about a normal model required for the t test to be valid and want to check this

condition. You recall that a QQ plot helps to assess if a population of responses can be considered normally

distributed. How many QQ plots do you need to make in this case?

One QQ Plot of the sample of differences in measurements

One QQ Plot of the population of differences in measurements

Two QQ Plots, one for the sample of measurements by Tech A and one for the sample of measurements by

Tech B

Two QQ Plots, one for the population of measurements by Tech A and one for the population of

measurements by Tech B

Question 10 Background :

Independent Samples or Paired?

Independent or Paired? For each scenario below determine if it is a paired data scenario or based on two

independent samples.

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Question 10 Subquestions

10.a

2 point(s)

A study was conducted to compare the grade point averages of sophomores who live in campus dormitories

with the grade point averages of sophomores who live off campus. Twenty sophomores were randomly selected

from campus dormitories at a college, and 20 other sophomores were randomly selected from students who live

off campus.

Paired Data

Two Independent Samples

10.b

0 point(s)

An experiment was performed to compare the reaction time to two types of traffic signs. One sign type is

Prohibitive (e.g., No Left Turn) and the other sign type is Permissive (e.g., Left Turn Only). Ten drivers were used

in the study. Each driver was presented with 40 traffic signs, 20 of each type, in a random order. For all 40 signs,

reaction time was recorded and the mean of each sign-type calculated.

Paired Data

Two Independent Samples

10.c

0.5 point(s)

A study was conducted to evaluate the average gain in strength due to a weight-training program. Thirteen

individuals each took a strength test both before and again after they participated in a six-week weight-training

program.

Paired Data

Two Independent Samples

Question 11 Background :

Which t-test?

Which t test? For each scenario below determine which type of hypothesis test (one-sample t test for a

population mean, paired-samples t test for the population mean difference, or independent-samples t test for the

difference in two population means) should be used to assess each of the following alternative hypotheses.

Question 11 Subquestions

11.a

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2 point(s)

On average, wives have more years of education than their corresponding husbands.

one-sample t test for a population mean

paired-samples t test for the population mean difference

independent-samples t test for the difference in two population means

11.b

0 point(s)

The average number of years of education for wives is 13.4 years.

one-sample t test for a population mean

paired-samples t test for the population mean difference

independent-samples t test for the difference in two population means

11.c

0.5 point(s)

The average age of persons marrying for the first time in the 1970s was lower than the average age of persons

marrying for the first time in the 2000s.

one-sample t test for a population mean

paired-samples t test for the population mean difference

independent-samples t test for the difference in two population means

Question 12 : 0.5 point(s)

What have you used?

Which of the following resources have you used during your time as a UM student to date?

Question 12 Multiple Choice Options

Office hours held by an instructor

Office hours held by a GSI

tutoring

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Edit Assignment (/#!/students/56d718ee422c530b007d5104/do)

56e8ab416f07640b00dd9f0a

peer-led study groups

None of the above