Math 140
Review for Exam 3 –Ch8, 9, & 10
Chapter 8
1.
Read the following problem description then write the correct null and alternative hypothesis. A new drug is being tested to see
whether it can reduce the rate of asthma attacks in children ages 5 to 14 with asthma ages. The rate of asthma attacks in the
population of concern is 0.0744.
2.
A researcher wishes to test the claim that the proportion of children with blue eyes in his region is different than one in six,
the national rate of blue eyes in children. State and explain the null and alternative hypothesis that should be used to test the
claim.
3.
A janitor at a large office building believes that his supply of light bulbs has more defective bulbs. The janitor's null hypothesis is
that the supply of light bulbs has a defect rate of p 0.07 (the light bulb manufacturer's stated defect rate). Suppose he does a
hypothesis test with a significance level of 0.05.
a)
Find the null and alternative hypothesis
b)
Suppose that the janitor tests 300 randomly selected light bulbs and finds that 27 bulbs are defective. What value of the test
statistic should he report? Round to the nearest hundredth.
4.
Suppose that the following is to be tested: H0 : p 0.35 and Ha : p 0.35 . Calculate the observed z-statistic for the following
sample data: Forty out of eighty test subjects have the characteristic of interest. Round to the nearest hundredth.
5.
Which of the following is not one of the four steps of the hypothesis test?
a. State the null and alternative hypothesis about the population parameter.
b. Make a decision to reject or not reject the null hypothesis.
c. State the level of significance, choose a test, and check the conditions for the test.
d. All of the above are steps of the hypothesis test.
6.
A researcher is wondering whether the drinking habits of adults in a certain region of the country are in the same proportion as
the general population of adults. The researcher decides to collect data from adults working in "blue collar" jobs to see whether
their drinking habits are in the same proportion as the general public. The null hypothesis for this test is H0 : p 0.26 and the
alternative hypothesis is Ha : p 0.26 . The researcher collected data from a random sample of 90 adults with "blue collar" jobs
of which 30 stated that they drank once a week or less in the last month. Assume that the conditions that must be met in order
for us to use the N0,1 distribution as the sampling distribution are satisfied.
a) Find the values of the sample proportion, the test statistic, and the p-value associated with the test statistic. Round all
values to the nearest thousandths.
b)
7.
8.
Based on a 5% significance level, write a conclusion by interpreting the p-value. Be sure to clearly state the decision
regarding the null hypothesis.
A researcher believes that the reading habits of men and women are different. He takes a random sample from each population
and records the response to the question, "Did you read at least one book last month?"
The null hypothesis is H0 : pwomen pmen Choose the correct alternative hypothesis.
a.
Ha : pwomen pmen
b.
Ha : pwomen pmen
c.
Ha : pwomen pmen
d.
Ha : p 0
The researcher believes that children who attend elementary school in a rural setting are more physically active then children
who attend elementary school in an urban setting. The researcher collects a random sample from each population and records
the proportion of children in each sample who reported participating in at least one hour of rigorous activity a day. The data is
summarized in the table below. Assume the all conditions for proceeding with a two-sample test have been met.
Rural
n1 90
Urban
n2 78
x1 74
x2 55
Find the z-statistic (rounded to the nearest hundredth) and p-value (rounded to the nearest thousandth) for this hypothesis test. Using
a 5% significance level, state the correct conclusion regarding the null hypothesis, H0 : prural purban .
a)
Find the percentage of children that reported participating in at least one hour of rigorous activity a day from each group.
Compare the percentages. Do the initial (untested) findings show what the researcher expected?
b)
State the null and alternative hypotheses
c)
Find the test statistic
d)
Find the P-value
e)
Conclusion:
9.
A researcher conducts a hypothesis test on a population proportion. Her null and alternative hypothesis are H0 : p 0.4
and Ha : p 0.4 . The test statistic and p-value for the test are z3.01 and p value 0.0013 . For a significance level of
0.05 , choose the correct conclusion regarding the null hypothesis.
a. There is not sufficient evidence to reject the null hypothesis that the population proportion is equal to 0.4.
b. There is sufficient evidence to accept the null hypothesis that the population proportion is equal to 0.4.
c. There is sufficient evidence to conclude that the population proportion is significantly different from 0.4.
d. There is not sufficient evidence to conclude that the population proportion is significantly different from 0.4.
10.
Read the following then choose the appropriate test and name the population(s).
A researcher asks random samples of men and women whether they had purchased organically grown food in the last three months.
He wants to determine whether the proportion of women who purchase organically grown food is greater than the proportion of men
who purchase organically grown food.
a)
b)
c)
d)
One-proportion z-test; the population is all men.
One-proportion z-test; the population is all women.
Two-proportion z-test; one population is all men and the other population is all women.
Two-proportion z-test; one population is all adults who buy organically grown food and the other population is all adults
who do not buy organically grown food.
11. Shade the approximate area under the Normal curve that would represent the p-value for the alternative hypothesis and z-score,
and then calculate the p-value. Round to the nearest thousandth.
The alternative hypothesis is a two-tailed with a z-score1.88
12. Two different students conduct a coin flipping experiment with a left- tailed alternative. The obtain the following test statistics:
Student #1: z2.05
Student #2: z1.28
Which of the test statistics has a smaller p-value and why?
13. The Harris Poll conducted a survey in which they asked “How many tattoos do you currently have on your body?” Of the 1,205
males surveyed, 181 responded that they had at least one tattoo. Of the 1,097 females surveyed, 143 responded that they had at
least one tattoo. At level of significance of 0.05, is the difference in proportions of females that have at least one tattoo different
from the proportion of males that have at least one tattoo.
a) State the hypotheses
b)
What kind of test do you have? (Right tailed-test, left tailed-test, or Two tailed-test)
c)
Find the test statistic
d)
Find the P-value
e)
Shade the approximate area under the Normal curve that would represent the p-value for the test statistics
f)
Conclusion:
Chapter 9
14. A sprint duathlon consists of a 5 km run, a 20 km bike , followed by another 5 km run. The mean finish time of all participants
in a recent large duathlon was 1.67 hours with a standard deviation of 0.25 hours. Suppose a random sample of 30 participants
was taken and the mean finishing time was found to be 1.59 hours with a standard deviation of 0.30 hours.
a) In this example, the numerical values of 1.67 hours and 0.25 hours represent statistic or parameter? Explain.
b)
Calculate the standard error for the mean finish time of 30 randomly selected male participants. Round your answer to
nearest thousandths.
c)
Suppose that he process of drawing samples of size 30 from the population of all male participants is repeated 100 times.
If possible, sketch and describe what the sampling distribution of the means will look like and state the approximate value
of the distribution.
15. Which of the following is not a true statement about the Central Limit Theorem for sample means?
a. All of the following statements are true about the Central Limit Theorem for sample means
b. If the sample size is large, it doesn't matter what the distribution of the population it was drawn from is, the normal
distribution can still be used to perform statistical inference.
c. If conditions are met, the mean of the sampling distribution is equal to the population mean.
d. The Central Limit Theorem helps us find probabilities for sample means when those means are based on a random
sample from a population.
16. Suppose that the average pop song length in America is 4 minutes with a standard deviation of 1.25 minutes. It is known
that song length is not normally distributed. Suppose a sample of 25 songs is taken from the population. What is the
approximate probability that the average song length will be less than 3.5 minutes? Round to the nearest thousandth.
17. Which of the following statements is not true about the t-distribution?
a. All of the following statements about the t-distribution are true.
b. For small sample sizes, the t-distribution has all the same properties of the normal curve.
c. Like the normal distribution, the t-distribution is symmetric and unimodal.
d. Since population standard deviation is usually unknown, the standard error uses the sample standard deviation to
estimate the population standard deviation.
18. Suppose that the average pop song length in America is 4 minutes with a standard deviation of 1.25 minutes. It is known that
song length is not normally distributed. Suppose a sample of 25 songs is taken from the population. What is the approximate
probability that the average song length will be less than 3.5 minutes? Round to the nearest thousandth.
19. The weights at birth of five randomly chosen baby giraffes were 111, 115, 120, 103, and 106 pounds. Assume the
distribution of weights is normally distributed. Find a 95% confidence interval for the mean weight of all baby giraffes.
Give the confidence interval in the form " estimate margin of error ". Round to the nearest tenth of pound.
20. Suppose a consumer product researcher wanted to find out whether a highlighter lasted longer than the manufacturer's claim that
their highlighters could write continuously for 14 hours. The researcher tested 40 highlighters and recorded the number of
continuous hours each highlighter wrote before drying up. Test the hypothesis that the highlighters wrote for more than 14
continuous hours. Following are the summary statistics:
x 14.5 hours, s 1.2 hours
Write the null and alternative hypothesis. Report the test statistic, p-value, your decision regarding the null hypothesis. At the
5% significance level, state your conclusion about the original claim. Round all values to the nearest thousandth.
a)
State the hypotheses
b)
What kind of test do you have? (Right tailed-test, left tailed-test, or Two tailed-test)
c)
Find the test statistic
d)
Find the P-value
e)
Conclusion:
21. An economist conducted a hypothesis test to test the claim that the average cost of eating a meal at home increased from 2009
to 2010. The average cost of eating a meal at home in 2009 was $5.25 per person per meal. Assume that all conditions for
testing have been met. He used technology to complete the hypothesis test. Following is his null and alternative hypothesis and
the output from his graphing calculator.
H0 : $5.25
Ha : $5.25
T-Test
 5.25
t 4.644687288
p 3.4015934 E 5
x 6.31
Sx 1.25
n 30
At the 5% significance level, write the statement explaining what his decision regarding the null hypothesis should be and
a statement summarizing his collusion regarding the claim.
22. State whether the situation has dependent or independent samples.
a) The reading level of a random sample of men and a random sample of women are measured. Researchers want know
whether women typically read at a higher level than men.
b)
The college GPA's of identical twins are compared to see whether the means are different.
23. A researcher wants to know whether athletic women are more flexible than non-athletic women. For this experiment, a
woman who exercised vigorously at least four times per week was considered "athletic". Flexibility is measured in inches on a
sit & reach box. Test the researcher's claim using the following summary statistics:
Athletic women
n 50
x 5.0 inches
s 1.4 inches
Non-athletic women
n 30
x 4.6 inches
s 0.8 inches
Assume that all conditions for testing have been met. Report the null and alternative hypothesis, test statistic and p-value.
At the 1% significance level, state your decision regarding the null hypothesis and your conclusion about the original claim.
Round all values to the nearest thousandth.
24. A researcher wants to know if mood is affected by music. She conducts a test on a sample of 4 randomly selected adults and
measures mood rating before and after being exposed to classical music. Test the hypothesis that mood rating improved after
being exposed to classical music. Following are the mood ratings for the four participants:
Participant #1
Participant #2
Participant #3
Participant #4
Before Exposure
4
5.2
3.2
4.5
After Exposure
5
6
4
6.2
Assume that all conditions for testing have been met. Report the null and alternative hypothesis and p- value. At the 5%
significance level, state your decision regarding the null hypothesis and your conclusion about the original claim. Round all
values to the nearest thousandth.
Chapter 10
25. The following table summarizes the outcomes of a study that researchers carried out to determine if females expressed a
greater fear of public speaking then males.
Expressed a fear of public speaking
Did not express a fear of public
speaking
Men
112
91
Women
79
82
a)
Describe the categorical variables that are summarized in the table. Describe one association that could be tested based on
the information from the table.
b)
Calculate the percentage of people in the study that did not express a fear of public speaking. Round to the nearest tenth of
a percent.
c)
Find the expected number of men who should express a fear of public speaking, if the variables are independent. Round to
the nearest hundredth.
26. Choose the statement that is not true about the chi- square distribution
a. Values for the chi-square statistic (on the horizontal axis) can be negative, positive, or zero.
b. The shape of the chi-square distribution depends on the degrees of freedom.
c. The lower the degrees of freedom, the more skewed to the right the chi-square distribution will be.
d. All of the above statements are true about the chi-square distribution.
27. Mars, manufacturer of M&Ms candies, used to publish the distributions of colors for M&Ms but no longer does that. Someone
gathered the colors for a large number of M&M and they are reported in the table. Test the hypothesis that there are equal
numbers in each color using a significance level of 0.05.
Color
Red
Green
Yellow
Orange
Blue
Brown
.
Number
516
669
462
754
795
426
a)
State the null and alternative hypothesis
b)
Find the test statistics and p-value
c)
Write the decision regarding the null hypothesis and conclusion statement
28. Suppose a researcher was interested in learning more about parents' concerns when their children start elementary
school. The researcher asks the parents of 800 randomly selected first graders in rural school district and the parents of
950 randomly selected first graders in an urban school district to rate their level of concern with the following
statement: We are (a) not at all concerned (b) somewhat concerned or (c) very concerned about the nutrition level of
school lunches. If we wanted to test whether there was an association between the response to the question and the
type of school district that the first grader was attending, would this be a test of homogeneity or of independence?
29. Suppose a random sample of 1,220 U. S. adults were asked about their opinion regarding federal spending on infrastructure
(i.e. roads and bridges). Respondents were asked whether federal spending on infrastructure was (a) too low, (b) adequate, or
(c) too high. Respondents were classified by income level. If we wanted to test whether there was an association between the
response to the question and income level, would this be a test of homogeneity or of independence?
30. Suppose a random sample of 1,220 U. S. adults were asked about their opinion regarding federal spending on infrastructure
(i.e. roads and bridges). Respondents were asked whether federal spending on infrastructure was (a) too low, (b) adequate,
or (c) too high. Respondents were classified by income level. State the correct hypotheses to test whether there is an
association between the response to the question and income level.
31. The table below shows the gender and the percentage of each gender that spent different amounts at a local
hardware store. The data was taken from a random sample of single shoppers collected over five consecutive
Saturdays at the hardware store. Choose the reason(s) why you cannot do a chi-square test with this data.
Male shopper
shopper
a.
b.
c.
d.
$0-$20
27%
46%
>$20-$50
42%
35%
>$50
31% Female
19%
There is not enough information to convert the percentages to counts.
The samples were not collected randomly
The data are from the entire population, not a sample, so inference is unnecessary.
All of the above.
32. Suppose a study was conducted to see whether there is an association between marital status and vehicle
color. The table below shows the results from the study. Assume all conditions for testing have been met.
Vehicle color
red
black/white/silver
blue/green/other
Married
45
22
19
Unmarried
35
11
18
This was an observational study of randomly chosen vehicle owners. Test the hypothesis that marital status and vehicle
color are associated, using a significance level of 0.05. State the correct decision regarding the null hypothesis and
conclusion. Refer to the computer output on the next page.
2 Test
X 2 1.754774169
p .4158681217
df=2
33. A health foods store owner is thinking about carrying some new products and is interested in her customer's opinions. The
shop owner decides to randomly sample 202 customers and ask them whether they have (a) heard about the health benefits of
coconut milk and (b) whether they have heard of the health benefits of Quinoa flour. She also asked each respondent how
often they typically make purchases during a month. The table below shows the results from the study. Assume all
conditions for testing have been met.
1-2 purchases/mo.
3+ purchases/mo.
5+ purchases/mo.
Coconut milk, yes?
57
21
11
Coconut milk, no?
12
11
10
Quinoa flour, yes?
19
25
10
Quinoa flour, no?
10
8
8
Test the hypothesis that how the respondents answered the questions is associated with number of monthly purchases, using a
significance level of 0.05. State the correct decision regarding the null hypothesis and conclusion. Refer to the computer output
below.
2 Test
X 2 19.43852896
p .0034836969
df=6