Problem Set Section 7.1The Basics of Hypothesis Testing

Problem Set Section 7.1The Basics of Hypothesis Testing

Null and Alternative Hypothesis: State the null and alternative hypothesis:
Coca-Cola distributes its product in 20 ounce bottles. Is the average weight of the cola in the bottles really equal to 20 ounces?
Among Americans eligible to vote does the percentage of those who favor overturning Roe v. Wade exceed 50%?
As part of an advertising campaign to attract farmers, the Iowa Farm Bureau claims that the state's precipitation amount averages more than two and one-half feet per year. Is their claim correct?
A company that has a 10% market share launches a marketing campaign. At the end of the campaign period the company conducts a survey in order to assess whether its market share has increased.
A bottling machine is set to automatically fill each bottle with 355 ml. of soft drink. To check whether the machine needs to be readjusted, a quality control inspector examines a random sample of newly filled bottles.
According the U.S Census Bureau, 11.8% of registered births in the United States in 2000 were to teenage mothers. A sociologist believes that this percentage has decreased since then.
Critical Value(s): Find the Critical Values. Assume that the normal distribution applies.
Two-tailed test; α = .05.
Right-tailed test; α = .1
α = .01; H1 is µ < 20.
α = .025; H1 is p ≠ 20.
P-Values: A two-tailed probability is .03. What is the one-tailed probability if the effect were in the specified direction? What would it be if the effect were in the other direction?
P-Values: Why would it be wrong according to the Classic view of hypothesis testing to write in a research report: "The effect was significant, p = .0082."
P-Values: Find the p-value if,
the test statistic in a right-tailed test is z = .055.
the test statistic in a two-tailed test is z = 1.95.
with H1: p < .12 the test statistic is z = -0.90.

Hypothesis Test Conclusion: Use the hypotheses in problem #1 above to state a conclusion based on the results of the test.
Suppose the null hypothesis is not rejected.
Suppose the null hypothesis is rejected.
Suppose the null hypothesis is not rejected.
Suppose the null hypothesis is rejected.
Suppose the null hypothesis is not rejected.
Suppose the null hypothesis is rejected.
Type I and Type II Errors: Identify the Type I and Type II errors
The mean score for all NBA games during a particular season was less than 100 points per game.
The mean age of bus drivers in Chicago is 54.5 years.
Type I and Type II Errors: Suppose that you are a lawyer that is trying to establish that a company has been unfair to minorities with regard to salary increases. Suppose the mean salary increase per year is 8%. You set the null hypothesis to be

H0: = .08

H1:  < .08

What is a type I error? Include context of the problem in your statement.
What is a type II error? Include the context of the problem in your statement.

Problem Set Section 7.2Testing a Claim About a

Population Mean: Large Samples

Determine whether z/2 can be used to test the hypothesis.
n = 50;  = 12; and the distribution is skewed.
n = 25;  = 12; and the distribution is skewed.
n = 25;  = 12; and the distribution is approximately normal.
n = 25; s = 12; and the distribution is approximately normal.
n = 150; s = 12; and the distribution is skewed.
Find the test statistic, p-value, critical values, and state the final conclusion.
Ho: µ ≤ 118; H1 µ > 118; n = 50;  = 12; α = .05
Ho: µ = 5; H1 µ ≠ 5; n = 80;  = 2.5; α = .01
Cholesterol of Children: Suppose we want to show that only children have an average higher cholesterol level than the national average. It is known that the mean cholesterol level for all Americans is 190. Construct the relevant hypothesis test: We test 100 only children and find that the sample mean and standard deviation is 198 and 15 respectively. Use a 0.1 significance level.
Insurance Rates: An insurance company is reviewing its current policy rates. When originally setting the rates they believed that the average claim amount was $1,800. They are concerned that the true mean is actually higher than this, because they could potentially lose a lot of money. They randomly select 40 claims, and calculate a sample mean of $1,950. Assuming that the standard deviation of claims is $500, and set α =.05, test to see if the insurance company should be concerned.
Mean Weight of Hamburgers: The Federal Trade Commission received a number of complaints that a hamburger chain is selling hamburgers promoted as weighing 1/4 lb. prior to cooking; but for the most part these hamburgers do not weight 1/4 lb. Before formally opening an investigation the FTC took a random sample of 100 such burgers. The average weight before cooking was .268 lb. Assume  = 0.09 lb. At the 1% level of significance is there sufficient evidence to warrant an investigation?

Mean Weight of Bottles of Ketchup: Bottles of ketchup are filled automatically by a machine that must be adjusted periodically to increase or decrease the average content per bottle. Each bottle is supposed to contain 18 oz. It is important to detect an average content significantly above or below 18 oz so that the machine can be adjusted, too much ketchup per bottle would be unprofitable, while too little would be a poor business practice and open the company up to law suites about invalid labeling. We select a random sample of 80 bottles filled by the machine and compute their average weight to be 18.2. Assume  = 0.7334. Should we adjust the machine? Use a comfort level of 5%. Use a Confidence interval to validate your conclusion. The Minitab display is shown below.

One-Sample Z

Test of mu = 18 vs not = 18

The assumed standard deviation = 0.7334

N Mean SE Mean 95% CI Z P

80 18.2000 0.0820 (18.0393, 18.3607) 2.44 0.015

Problem Set Section 7.3Testing a Claim About a

Population Mean: Small Samples

Determine whether z/2, t/2 or neither can be used to test the hypothesis.
n = 50; s = 12; and the distribution is skewed.
n = 25; s = 12; and the distribution is skewed.
n = 25;  = 12; and the distribution is approximately normal.
n = 25; s = 12; and the distribution is approximately normal.
n = 150; s = 12; and the distribution is skewed.

2.Find the test statistic, p-value, critical values, and state the final conclusion.

Ho: µ ≤ 118; H1 µ > 118; n = 25; s = 12; α = .05
Ho: µ = 5; H1 µ ≠ 5; n = 15; s = 2.5; α = .01

Is the temperature required to damage a computer on the average less than 110 degrees? Because of the price of testing, twenty computers were tested to see what minimum temperature will damage the computer. The damaging temperature averaged 109 degrees with a standard deviation of 3 degrees. It is determined that the temperatures are approximately normally distributed. (use 
The expense of moving the storage yard for the Consolidated Delivery Service is justified only if it can be shown that the mean travel distance will be less than 214 miles. In trial runs of 12 delivery trucks, the mean and standard deviation are found to be 198 miles and 42 miles respectively. At the 0.01 level of significance, test the claim that the mean is less than 214 miles. Assume the distances are approximately normally distributed. Should the storage yard be moved?
Age of Death Row Inmates: IN 2002, the mean age of an inmate on death row was 40.7 years, according tot the U.S. Department of justice. A sociologist wondered whether the mean age of a death-row inmate has changed since then. She randomly selected 25 death-row inmates and find that their mean age is 38.9, with a standard deviation of 9.6.
Do you believe the mean age has changed? Use a 0.02 level of significance.
Construct a 98% confidence interval about the mean age. What does the interval imply?

Driving to Campus: Trying to encourage people to stop driving to campus, the university claims that on average it takes people 30 minutes to find a parking space on campus. A student doesn’t think it takes so long to find a spot. In fact he has a sample of the last five times he drove to campus. He calculated mean time to be 20 and the standard deviation to be 6. Assuming that the time it takes to find a parking spot is normal, perform a hypothesis test with significance level = .10 to see if his claim is correct. The Minitab output is displayed below.

Test of mu = 30 vs < 30

90% Upper

N Mean StDev SE Mean Bound T P

5 20.00 6.00 2.68 24.11 -3.73 0.010

Sprinkler Activation: The times of first sprinkler activation for a series of tests with fire prevention sprinkler systems using an film-forming foam were (in sec)

27 41 22 27 23 35 30 33 24 27 28 22 24

The system has been designed so that true average activation time is at most 25 sec under such conditions. Does the data strongly contradict the validity of this design specification? Test the relevant hypothesis at significance level .05. The Minitab output is displayed below. Comment on the whether the data appears to be normal.

Test of mu = 25 vs > 25

Variable N Mean StDev SE Mean T P

Times 13 27.92 5.62 1.56 1.88 0.043

Conforming Gold Balls: Te USGA requires that golf balls have a diameter that is 1.68 inches. To determine if Maxfli XS golf balls conform to USGA standards, a random sample of Maxfli XS golf balls was selected. The diameters are shown below.

Verify the data appear to be normally distributed?
Do the balls conform to USGA standards? Use a 0.01 level of significance.
Determine and interpret the p-value. Use your calculator.

Problem Set Section 7.4Testing a Claim About a

Population Proportion

Fair Coins: Suppose in the first 50 tosses of a coin there were 20 heads and 30 tails. Would this suggest the coin was not balanced?
Fair Die: I roll a single die 1,000 times and obtain a “6” on 204 rolls. Is there significant evidence to suggest that the die is not fair? (That is, if p is the proportion of the rolls that come up 6 is it reasonable to assume that p = 1/6?).
Voting for Mayor: Candidate Jones is one of two candidates running for mayor of Central City. A random polling of 672 registered voters finds that 323 (48% of those polled) will vote for him. It is reasonable to assume that the race for mayor is a tossup? (That is, if p is the proportion of the population who will vote for Jones, is it reasonable to assume that p = 0.5?).
Voting Democrat: Suppose that you interview 1000 exiting voters about who they voted for governor. Of the 1000 voters, 550 reported that they voted for the democratic candidate. Is there sufficient evidence to suggest that the democratic candidate will win the election at the .01 level?
Executive Traveler’s Club: A plan for an executive traveler’s club has been developed by an airline on the premise that 5% of its current customers would qualify for membership. A random sample of 500 customers yielded 40 who would qualify. Using this data, test at level .01 the null hypothesis that the company’s premise is correct against the alternative that it is not correct.
Beetle Infestation: 1500 randomly selected pine trees in the Lake Tahoe National Forest were tested for traces of the Bark Beetle infestation. It was found that 153 of the trees showed such traces. Test the hypothesis that more than 10% of the Tahoe trees have been infested. (Use a 5% level of significance)
Chromosome Defect: The Incidence of a certain type of chromosome defect in the U.S. adult male population is believed to be 1 in 75. A random sample of 800 individuals in U.S. penal institutions reveals 16 who have such defects. Can it be concluded that the incidence rate of this defect among prisoners differs from the presumed rate for the entire adult male population? Use a 2% level of significance.

Tattoos: In 2001, 23% of American university undergraduate student had at least one tattoo. A health practitioner obtains a random sample of 1026 university undergraduate and finds that 254 have at least one tattoo. Has the proportion of American university undergraduate students with at least one tattoo changed since 2001? The Minitab out is listed below. Use a 10% level of significance.

Test and CI for One Proportion

Test of p = 0.23 vs p not = 0.23

Exact

Sample X N Sample p 90% CI P-Value

1 254 1026 0.2476 (0.22540, 0.26970) 0.1813

Problem Set Section 7.5Testing a Claim About a

Population Variance

Find the critical value(s) for chi square that correspond to the given information.
H1:  ≠ 12, α = .05 ; n = 12 ; s=10
H1:  > 12, α = .01 ; n = 5 ; s=18
A local bank needs information concerting the standard deviation of the checking account balances of its customers. From previous information it was assumed to be $250. A random sample of 61 accounts was checked. The standard deviation was $286.20. At α=.01, test the bank’s assumption. Assume that the account balances are normally distributed. Interpret.

Selected Answers:

Section 7.1

1b) Ho: p≤0.5, H1: p>0.51e) Ho:μ=355, H1:µ≠355

3) 0.015, same

4) Must state the value of α to compare to the p-value

5a) 0.47815b) 0.05125c) 0.1841

7a) Type I: support μ<100 when µ≥100 is true

Type II: fail to support μ<100 when µ≥100 is not true

8a) We put sanctions on the company when they were not being discriminatory

8b) We allow the company to go about its discriminatory ways.

Section 7.2

1a) yes1b) no1c) yes 1d) no 1e) yes

3) z=5.33, CV=1.28, reject5) z=2.0, CV=2.575, fail to reject

Section 7.3

1a) t or z 1b) neither 1c) z1d) t1e) t or z

2a) t=0.833, p-val=0.2064, cv=1.711, fail to reject

4) t=-1.31, CV=2.718, fail to reject, no

Section 7.4

4) z=3.15, cv=2.326, reject, yes

6)z=0.258, cv=1.645, fail to reject,

Section 7.5

2) critical values 35.534 & 91.952 ; test statistic 78.634 ; fail to reject Ho: