ECMT1010 Online homework 5 | Complete Solution

Chapter 4, Section 3, Exercise 075

Match the -values with the appropriate conclusion:

(a) The evidence against the null hypothesis is significant, but only at the level.

 0.000130.62510.02190.0883

(b) The evidence against the null and in favor of the alternative is very strong.

 0.62510.02190.000130.0883

(c) There is not enough evidence to reject the null hypothesis, even at the level.

 0.000130.02190.08830.6251

(d) The result is significant at a level but not at a level.

 0.08830.62510.000130.0219
Chapter 4, Section 3, Exercise 082
Chapter 4, Section 3, Exercise 082 Sleep or Caffeine for Memory?

The consumption of caffeine to benefit alertness is a common activity practiced by of adults in North America. Often caffeine is used in order to replace the need for sleep. One recent study1 compares students' ability to recall memorized information after either the consumption of caffeine or a brief sleep. A random sample of adults (between the ages of ) were randomly divided into three groups and verbally given a list of words to memorize. During a break, one of the groups takes a nap for an hour and a half, another group is kept awake and then given a caffeine pill an hour prior to testing, and the third group is given a placebo. The response variable of interest is the number of words participants are able to recall following the break. The summary statistics for the three groups are in the table below. We are interested in testing whether there is evidence of a difference in average recall ability between any two of the treatments. Thus we have three possible tests between different pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo.

Group     Sample size     Mean     Standard Deviation
Sleep             
Caffeine             
Placebo             

1 Mednick, Cai, Kanady, and Drummond, "Comparing the benefits of caffeine, naps and placebo on verbal, motor and perceptual memory", Behavioural Brain Research, 193 (2008), 79-86.

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(a) In the test comparing the sleep group to the caffeine group, the -value is .

What is the conclusion of the test?

 RejectDo not reject .

In the sample , which group had better recall ability?

 SleepCaffeine

According to the test results, do you think sleep is really better than caffeine for recall ability?

 YesNo

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(b) In the test comparing the sleep group to the placebo group, the -value is .

What is the conclusion of the test, using a significance level?

 RejectDo not reject .

What is the conclusion of the test, if we use a significance level?

 RejectDo not reject .

How strong is the evidence of a difference in mean recall ability between these two treatments?

 No evidence at allOnly moderately strongVery strong

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(c) In the test comparing the caffeine group to the placebo group, the -value is .

What is the conclusion of the test?

 RejectDo not reject .

In the sample , which group had better recall ability?

 CaffeinePlacebo

According to the test results, would we be justified in concluding that caffeine impairs recall ability?

 YesNo

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(d) According to this study, what should you do before an exam that asks you to recall information?

Take a nap.
    

Have a placebo.
    

Have some coffee.

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Chapter 4, Section 4, Exercise 128
Chapter 4, Section 4, Exercise 128 Arsenic in Chicken

A restaurant chain is measuring the levels of arsenic in chicken from its suppliers. The question is whether there is evidence that the mean level of arsenic is greater than ppb, so we are testing vs , where represents the average level of arsenic in all chicken from a certain supplier. It takes money and time to test for arsenic so samples are often small. Suppose chickens from one supplier are tested, and the level of arsenic (in ppb) are .

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(a) What is the sample mean for the data?

Round your answer to the nearest integer.

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(b) Translate the original sample data by the appropriate amount to create a new data set in which the null hypothesis is true. How do the sample size and standard deviation of this new data set compare to the sample size and standard deviation of the original data set?

They are different.
    

The sample size is the same but the standard deviation is different.
    

The standard deviation is the same but the sample size is different.
    

They are the same.

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(c) Write the six new data values from part (b) on six cards. Sample from these cards with replacement to generate one randomization sample. (Select a card at random, record the value, put it back, select another at random, until you have a sample size of , to match the original sample size.) Give the sample mean.

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the tolerance is +/-2%
Chapter 4, Section 5, Exercise 149
Chapter 4, Section 5, Exercise 149

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample, and give the significance level used. In addition, in each case for which the results are significant, state which group ( or ) has the larger mean.

Hypotheses: vs

(a) confidence interval for : to

Conclusion:  RejectDo not reject

Significance level:  95%5%90%99%1%

Group with the larger mean:  Group 1Group 2Cannot determine

(b) confidence interval for : to

Conclusion:  RejectDo not reject

Significance level:  99%1%90%5%95%

Group with the larger mean:  Group 1Group 2Cannot determine

(c) confidence interval for : to

Conclusion:  RejectDo not reject

Significance level:  90%5%95%1%99%

Group with the larger mean:  Cannot determineGroup 1Group 2
Chapter 4, Section 5, Exercise 156
Chapter 4, Section 5, Exercise 156

Are you "In a Relationship"?

A new study1 shows that relationship status on Facebook matters to couples. The study included college-age heterosexual couples who had been in a relationship for an average of months. In of the couples, both partners reported being in a relationship on Facebook. In of the couples, both partners showed their dating partner in their Facebook profile picture. Men were somewhat more likely to include their partner in the picture than vice versa. However, the study states: "Females' indication that they are in a relationship was not as important to their male partners compared with how females felt about male partners indicating they are in a relationship." Using a population of college-age heterosexual couples who have been in a relationhip for an average of months:

(a) A confidence interval for the proportion with both partners reporting being in a relationship on Facebook is about to . What is the conclusion in a hypothesis test to see if the proportion is different from ? What significance level is being used?

Conclusion:  RejectDo not reject

Significance level:  95%1%90%99%5%
(b) A confidence interval for the proportion with both partners showing their dating partner in their Facebook profile picture is about to . What is the conclusion in a hypothesis test to see if the proportion is different from ? What significance level is being used?

Conclusion:  RejectDo not reject

Significance level:  5%90%95%99%1%
1 Roan, Shari, "The true meaning of Facebook's 'in a relationship'", Los Angeles Times, February 23, 2012, reporting on a study in Cyberpsychology, Behavior, and Social Networking.
Chapter 5, Section 1, Exercise 013
Chapter 5, Section 1, Exercise 013 Find the specified areas for a normal density.

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(a) The area below on a distribution

Round your answer to three decimal places.

area

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the tolerance is +/-2%

(b) The area above on a distribution

Round your answer to three decimal places.

area

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(c) The area between and on a distribution

Round your answer to three decimal places.

area

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Chapter 5, Section 1, Exercise 020
Chapter 5, Section 1, Exercise 020 Find endpoint(s) on the given normal density curve with the given property.

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(a) The area to the left of the endpoint on a curve is about .

Round your answer to two decimal places.

endpoint

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(b) The area to the right of the endpoint on a curve is about .

Round your answer to the nearest integer.

endpoint

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Chapter 5, Section 1, Exercise 028
Chapter 5, Section 1, Exercise 028

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Choose the graph for the middle for a standard normal distribution.

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Choose the graph for the the middle for a standard normal distribution converted to a distribution.

    
    
    
    
    
Chapter 5, Section 1, Exercise 031
Chapter 5, Section 1, Exercise 031

Random Samples of College Degree Proportions

The distribution of sample proportions of US adults with a college degree for random samples of size is . How often will such samples have a proportion, , that is more than ?

Round your answer to one decimal place.

% of samples of 500 US adults will contain more than 30.0% with at least a bachelor’s degree.
Chapter 5, Section 1, Exercise 040
Chapter 5, Section 1, Exercise 040

Curving Grades on an Exam

A statistics instructor designed an exam so that the grades would be roughly normally distributed with mean and standard deviation . Unfortunately, a fire alarm with ten minutes to go in the exam made it difficult for some students to finish. When the instructor graded the exams, he found they were roughly normally distributed, but the mean grade was and the standard deviation was . To be fair, he decides to ‘‘curve” the scores to match the desired distribution. To do this, he standardizes the actual scores to -scores using the distribution and then ‘‘unstandardizes” those -scores to shift to .

What is the new grade assigned for a student whose original score was ?

Round your answer to the nearest integer.

new score

How about a student who originally scores an ?

Round your answer to the nearest integer.

new score
Chapter 5, Section 2, Exercise 044
Chapter 5, Section 2, Exercise 044

Find the indicated confidence interval. Assume the standard error comes from a bootstrap distribution that is approximately normally distributed.

A confidence interval for a proportion if the sample has with , and the standard error is

Round your answers to three decimal places.

The confidence interval is to .
Chapter 5, Section 2, Exercise 046
Chapter 5, Section 2, Exercise 046

Find the indicated confidence interval. Assume the standard error comes from a bootstrap distribution that is approximately normally distributed.

A confidence interval for a mean if the sample has with and , and the standard error is

Round your answers to three decimal places.

The confidence interval is to .
Chapter 5, Section 2, Exercise 059
Chapter 5, Section 2, Exercise 059

Where Is the Best Seat on the Plane?

A survey of air travelers1 found that prefer a window seat. The sample size is large enough to use the normal distribution, and a bootstrap distribution shows that the standard error is .

Use a normal distribution to find a confidence interval for the proportion of air travelers who prefer a window seat.

Round your answers to three decimal places.

The confidence interval is to .

1Willingham, A., ‘‘And the best seat on a plane is... 6A!,” HLNtv.com, April 25, 2012.
Chapter 5, Section 2, Exercise 061
Chapter 5, Section 2, Exercise 061 Smoke-Free Legislation and Asthma

Hospital admissions for asthma in children younger than years was studied1 in Scotland both before and after comprehensive smoke-free legislation was passed in March 2006. Monthly records were kept of the annualized percent change in asthma admissions, both before and after the legislation was passed. For the sample studied, before the legislation, admissions for asthma were increasing at a mean rate of per year. The standard error for this estimate is per year. After the legislation, admissions were decreasing at a mean rate of per year, with a standard error for this mean of . In both cases, the sample size is large enough to use a normal distribution.

1Mackay, D., et. al., ‘‘Smoke-free Legislation and Hospitalizations for Childhood Asthma,” The New England Journal of Medicine, September 16, 2010; 363(12):1139-45.

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(a) Find a confidence interval for the mean annual percent rate of change in childhood asthma hospital admissions in Scotland before the smoke-free legislation.

Round your answers to one decimal place.

The confidence interval is to .

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(b) Find a confidence interval for the same quantity after the legislation.

Round your answers to one decimal place.

The confidence interval is to .

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(c) Is this an experiment or an observational study?

    Experiment
    Observational study

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(d) The evidence is quite compelling. Can we conclude cause and effect?

    Yes
    No
Chapter 5, Section 2, Exercise 064
Chapter 5, Section 2, Exercise 064 Penalty Shots in World Cup Soccer

A study1 of penalty shots in World Cup Finals games between and found that the goalkeeper correctly guessed the direction of the kick only of the time. The article notes that this is ‘‘slightly worse than random chance.” We use these data as a sample of all World Cup penalty shots ever. Test at a significance level to see whether there is evidence that the percent guessed correctly is less than . The sample size is large enough to use the normal distribution. The standard error from a randomization distribution under the null hypothesis is .

1St.John, A., ‘‘Physics of a World Cup Penalty-Kick Shootout - 2010 World Cup Penalty Kicks,” Popular Mechanics, June 14, 2010.

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State the null and alternative hypotheses.

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What is the test statistic?

Round your answer to two decimal places.

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the tolerance is +/-2%

What is the -value?

Round your answer to three decimal places.

-value

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What is the conclusion?

    Reject and find evidence that the proportion guessed correctly is not less than half.
    Reject and find evidence that the proportion guessed correctly is less than half.
    Do not reject and do not find evidence that the proportion guessed correctly is less than half.
    Do not reject and find evidence that the proportion guessed correctly is less than half.

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Chapter 5, Section 2, Exercise 065
Chapter 5, Section 2, Exercise 065 How Often Do You Use Cash?

In a survey1 of American adults conducted in April 2012, reported having gone through an entire week without paying for anything in cash. Test to see if this sample provides evidence that the proportion of all American adults going a week without paying cash is greater than . Use the fact that a randomization distribution is approximately normally distributed with a standard error of . Show all details of the test and use a significance level.

1‘‘43% Have Gone Through a Week Without Paying Cash,” Rasmussen Reports, April 11, 2011.

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State the null and alternative hypotheses.

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What is the test statistic?

Round your answer to two decimal places.

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the tolerance is +/-2%

What is the -value?

Round your answer to two decimal places.

-value

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the tolerance is +/-2%

    
    
What is the conclusion?

    Do not reject and find evidence that the proportion is greater than .
    Do not reject and do not find evidence that the proportion is greater than .
    Reject and find evidence that the proportion is not greater than .
    Reject and find evidence that the proportion is greater than .