Chapter 6 Course Pack - MATH 1340
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MATH 1340 Chapter 6: Hypothesis Tests 1 of 8 Section 6.1: Basics of Hypothesis Tests Ex 1) State the null and alternative hypotheses in a statistical hypothesis test involving the given claim. Then determine if the test is left-tailed, right-tailed, or two-tailed. (a)
A restaurant claims that at least 80% of the items on its menu are made only from certified organic ingredients. (b)
The President claims that the average age of people collecting unemployment benefits is 43.8 years. (c)
A store claims that its customers save an average of more than $10 for every $100 they spend at their store compared to a competitor. Ex 2) Assume that a hypothesis test is to be conducted, and the test statistic is z
or t
(so the sampling distribution is bell-shaped). (a)
If a left-tailed test in conducted at the 0.01 significance level, sketch the sampling distribution of the test statistic and label the shaded area above the rejection region. (b)
If a two-tailed test with
= 0.1 is conducted, sketch the sampling distribution of the test statistic and label the shaded area above the rejection region. (c)
If the rejection region of a test is under both tails of the test statistic’s density curve, and the area under each tail that is above the corresponding rejection region is 0.02, sketch the sampling distribution of the test statistic and label the shaded area above the rejection region. What is the significance level of this test? Ex 3) A hypothesis test of the following claim is to be conducted: “the average time that a customer waits for his/her food to be served after placing an order is less than 15 minutes.” A random sample of 47 customer orders is selected, and it yields a test statistic of t
= -1.31. A significance level of 0.05 is to be used. (a)
Conduct this hypothesis test using the critical value method.
(b)
Conduct this hypothesis test using the P
-value method.
MATH 1340 Chapter 6: Hypothesis Tests 2 of 8 (c)
State the final conclusion of this hypothesis test in nontechnical language that addresses the original claim. (d)
If the true population mean waiting time is 14.2 minutes, then identify whether a type I error, a type II error, or a correct decision was made. Ex 4) A hypothesis test of the following claim is to be conducted: “the average attack percentage per game for the Monarchs’ volleyball team is 0.297.” (“Attack percentage” in volleyball is the number of kills minus the number of hitting errors, all divided by the total number of attacks). A random sample of 35 of their games is selected, and it yields a test statistic of t
= 2.27. A significance level of 0.1 is to be used. (a)
Conduct this hypothesis test using the critical value method.
(b)
Conduct this hypothesis test using the P
-value method.
(c)
State the final conclusion of this hypothesis test in nontechnical language that addresses the original claim. (d)
If the true average attack percentage per game is 0.305, then identify whether a type I error, a type II error, or a correct decision was made. Ex 5) A hypothesis test of the following claim is to be conducted: “the percentage of currently enrolled students who are required to take Elementary Statistics is more than 75%.” A random sample of 101 students is selected, and it yields a test statistic of z = 2.30. A significance level of 0.02 is to be used. (a)
Conduct this hypothesis test using the critical value method.
(b)
Conduct this hypothesis test using the P
-value method.
MATH 1340 Chapter 6: Hypothesis Tests 3 of 8 (c)
State the final conclusion of this hypothesis test in nontechnical language that addresses the original claim. (d)
If the true percentage of students required to take Elementary Statistics is 73.6%, then identify whether a type I error, a type II error, or a correct decision was made. Section 6.2: Hypothesis Tests Involving a Single Proportion Ex 6) In a random sample of 50 registered voters from Pleasantville, it was found that 27 of them voted in the previous city council election. The mayor of Pleasantville claims that at least 60% of the city’s registered voters actually did vote in the election. What is the value of the test statistic that would be used in a test of this claim? Ex 7) A survey of 561 customers of Save-Mart revealed that 58% of them have used the store’s mobile app at least once. The manager of the IT department for Save-Mart claims that more than 55% of customers have used the app. Find the value of the test statistic that would be use in a test of this claim. Ex 8) A geneticist claimed that less than 7.5% of fish given a specific growth hormone gene would incorporate it into their DNA. Scientists injected this gene into thousands of fish eggs. Of the 400 fish that grew from these eggs, 20 of them incorporated the gene into their DNA. Test the geneticist’s claim at the 0.05 significance level.
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MATH 1340 Chapter 6: Hypothesis Tests 4 of 8 Ex 9) A drug company claims that its new mosquito repellent spray is so effective that fewer than 1 in 4 people using this spray will get bitten. An investigative reporter wants to check the company’s claim. The reporter selects a random sample of 63 volunteers. Each person in this sample had this spray applied to their skin and then stood still in a small room with several mosquitoes for a full minute. This was repeated for each of the 63 people in the random sample. The reporter found that 24% of them had mosquito bites. Test the company’s claim at the 0.01 level of significance. Ex 10) The governor claims that the unemployment rate in Michigan is 7.3%. A random sample of 1274 people in Michigan’s labor force is obtained, and 8% of them are unemployed. Test the governor’s claim at the 0.05 significance level. Section 6.3: Hypothesis Tests Involving a Single Mean Ex 11) Find the test statistic used in a test where the null hypothesis is “
H
0
:
= 62” and a random sample of size 50
n
has a mean of 65.1
x
and standard deviation of 3.2
s
, and the population standard deviation is known to be 2.6
.
MATH 1340 Chapter 6: Hypothesis Tests 5 of 8 Ex 12) Find the test statistic used in a test where the null hypothesis is “
H
0
:
= 35” and a random sample of size 40
n
has a mean of 34.6
x
and standard deviation of 1.8
s
. Ex 13) In a random sample of 20 homeowners in Springfield, the average monthly electric bill during June was $64.16 and the standard deviation was $11.30. Assume that electric bills are approximately normally distributed. Test the claim that the average monthly electric bill for homeowners in Springfield is less than $70. Perform the test at the 0.05 and the 0.01 significance levels. Ex 14) A marketing firm wants to test the claim that the average age of consumers who shop at a particular store is 35. They collect a random sample of 64 shoppers at this store, and find that the mean is 36.2 years and standard deviation is 8.6 years. Conduct a hypothesis test of the claim at the 0.10 significance level.
MATH 1340 Chapter 6: Hypothesis Tests 6 of 8 Ex 15) Experts recommend that children’s screen time (including television, smartphones, tablets, and computers) should be limited to 2 hours or less per day. (
http://www.mayoclinic.org/healthy-lifestyle/childrens-health/in-depth/children-and-tv/art-20047952
, accessed 6/17/2015)
A sample of 129 randomly selected American children is selected, and their daily mean amount of screen time was found to be 2.6 hours, with a standard deviation of 0.9 hours. At the 0.005 significance level, test the claim that the average amount of screen time for American children is greater than the recommended 2-hour limit. Section 6.4: Hypothesis Tests from Two Samples Ex 16) Psychologists believe that people’s spending habits differ when they have larger denominations of money with them. In one particular study, students were randomly assigned to one of two groups, and both were given a total of $5. The 40 students in Group A were given a $5-bill, and the 34 students in Group B were given five $1-bills. All students from both groups were given the choice of keeping the money or buying snacks (gum, candy bars, and/or mints). In Group A there were 11 students who spent their $5-bill, and in Group B there were 18 students who spent their five $1-bills. Test the claim that the proportion of people who spend a $5-bill is less than the proportion of people who spend five $1-bills. Use a 0.05 level of significance.
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MATH 1340 Chapter 6: Hypothesis Tests 7 of 8 Ex 17) At the 0.05 significance level, test the claim that the two population means are the same if two independent samples from these populations produced the following statistics. Assume 1
2.3
and 2
1.7
. Sample size Sample mean Sample St. Dev. Sample from Population1 1
40
n
1
15.3
x
1
2.7
s
Sample from Population 2 2
51
n
2
14.5
x
2
1.6
s
Ex 18) A study was conducted to determine if there was any difference in grades between online classes that used proctored tests vs. online classes that only used online tests. Group 1 consisted of students who completed an online Intermediate Algebra class that had some of the tests proctored at a testing site on campus (these tests were closed-book and closed-
notes). Group 2 consisted of students who completed the same course, but these students were in a section of the class that had all tests online (so students could take the tests with their books and notes). The results from the first test in which Group 1 took a proctored test are below. n mean st. dev. Group 1 (proctored) 30 74.30 12.87 Group 2 (online) 32 88.62 22.09 Test the claim that the average score on proctored tests is less than the average score on (unproctored) online tests. Use a 0.025 level of significance, and assume that both populations of test scores are normally distributed. (Data from: Analysis of Proctored versus Non-
proctored Tests in Online Algebra Courses, by Michael Flesch and Elliot Ostler, MathAMATYC Educator
, Vol. 2, No. 1, August 2010, pp. 8-14)
MATH 1340 Chapter 6: Hypothesis Tests 8 of 8 Ex 19) Randomly selected respondents rated how wrong they believed animal research is by using a 5-point scale (where “0” = not wrong at all, “5” = completely wrong, and responses could be any real number between 0 and 5, inclusive). Researchers broke the responses into two groups by gender, and the results are summarized below. n mean st. dev. Females 25 4.97 1.57 Males 27 3.26 1.49 Assume that the population of responses from males and females are both normally distributed and they have the same population standard deviation. At the 0.005 significance level, test the claim that the mean response score from females is greater than the mean response score from males. Ex 20) A survey was conducted on the proposed retail pricing of a new smartphone. A random sample of 8 married couples is selected. After being shown a short video that describes the features of the smartphone, the husbands and wives each respond to a question that asks what the maximum amount is that they would be willing to pay for it. The results are in the following table. Test the claim that the mean difference of the husbands’ amount minus the wives’ amount is positive. Use a significance level of 0.025 and assume that the population of differences is approximately normal. Husbands Wives $600 $475 $499 $480 $550 $375 $650 $550 $500 $550 $475 $550 $700 $600 $625 $575