chapter 7 Review quiz
Quarters are currently minted with weights normally distributed and having a standard deviation of 0.063.New equipment is being tested in an attempt to improve quality by reducing variation. A simple random sample of 20 quarters is obtained from those manufactured with the new equipment, and this sample has a standard deviation of 0.041. Use a 0.025significance level to test the claim that quarters manufactured with the new equipment have weights with a standard deviation less than 0.063.Does the new equipment appear to be effective in reducing the variation of weights?
Reject the null hypothesis. A. Since the null hypothesis is rejected, the new equipment appears to be more effective.
Find the critical value(s) and rejection region(s) for a right-tailed chi-square test with a sample size n=18 and level of significance α=0.05.
critical value is 27.587
A local chess club claims that the length of time to play a game has a mean of more than 46 minutes. Write sentences describing type I and type II errors for a hypothesis test of this claim.
A type I error will occur if the actual mean of the length of time to play a game is less than or equal to 46 minutes, but you reject the null hypothesis, Upper H 0 : mu less than or equals 46H0: μ≤46. A type II error will occur if the actual mean of the length of time to play a game is greater than 46 minutes, but you fail to reject the null hypothesis, Upper H 0 : mu less than or equals 46 H0: μ≤46.
Describe type I and type II errors for a hypothesis test of the indicated claim. A lumber store claims that no more than 25% of its new customers will return to buy their next order of lumber.
A type I error will occur when the actual proportion of new customers who return to buy their next order of lumber is no more than 0.25, but you reject H0: p≤0.25.A type II error will occur when the actual proportion of new customers who return to buy their next order of lumber is more than 0.25, but you fail to reject H0:p≤0.25.
Decide whether the normal sampling distribution can be used. If it can be used, test the claim about the population proportion p at the given level of significance α using the given sample statistics. Claim: p<0.14; α=0.01; Sample statistics: p=0.12, n=25
No, because np is less than 5.The test cannot be performed.
Can a critical value for the χ2-test be negative? Explain.
No, in a χ2-distribution, all χ2-values are greater than or equal to 0.
Decide whether the normal sampling distribution can be used. If it can be used, test the claim about the population proportion p at the given level of significance α using the given sample statistics. Claim: p≠0.26; α=0.10; Sample statistics: p=0.19, n=100
Can the normal sampling distribution be used?Yes, because both np and nq are greater than or equal to 5.The critical value(s) are −1.64,1.64. z=-1.60 Fail to reject H0. The data do not provide sufficient evidence to support the claim.
What are the two types of hypotheses used in a hypothesis test? How are they related?
Null and alternative, they are complements.
Determine whether the claim stated below represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that (a) rejects the null hypothesis or (b) fails to reject the null hypothesis? A scientist claims that the mean incubation period for the eggs of a species of bird is less than 54 days.
Does the claim represent the null hypothesis or the alternative hypothesis? Since the claim does not contain a statement of equality, it represents the alternative hypothesis. Part 2 (a) How should you interpret a decision that rejects the null hypothesis? There is sufficient evidence to support the claim that the mean incubation period for the eggs of a species of bird is less than 54 days. Part 3 (b) How should you interpret a decision that fails to reject the null hypothesis? There is insufficient evidence to support the claim that the mean incubation period for the eggs of a species of bird is less than 54 days.
Determine whether the claim stated below represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that (a) rejects the null hypothesis or (b) fails to reject the null hypothesis? A researcher claims that the standard deviation of the life of a certain type of lawn mower is more than 3.6 years.
Does the claim represent the null hypothesis or the alternative hypothesis? Since the claim does not contain a statement of equality, it represents the alternative hypothesis. Part 2 (a) How should you interpret a decision that rejects the null hypothesis? There is sufficient evidence to support the claim that the standard deviation of the life of a certain type of lawn mower is more than 3.6 years Part 3 (b) How should you interpret a decision that fails to reject the null hypothesis? There is insufficient evidence to support the claim that the standard deviation of the life of a certain type of lawn mower is more than 3.6 years
The P-value for a hypothesis test is shown. Use the P-value to decide whether to reject H0 when the level of significance is (a) α=0.01, (b) α=0.05, and (c) α=0.10. P=
Fail to reject H0 because the P-value, 0.0495, is greater than α=0.01. Reject H0 because the P-value, 0.0495, is less than α=0.05. Reject H0 because the P-value, 0.0495, is less than α=0.10.
Match each P-value with the graph that displays its area without performing any calculations. Explain your reasoning. P=0.0158 and P=0.2946.
Graph (a) displays the area for P=0.0158 and graph (b) displays the area for P=0.2946 because the P-value is equal to the shaded area.
Explain how to find the critical values for a t-distribution.
Identify the level of significance α and the degrees of freedom, d.f.=n−1. If the hypothesis test is left-tailed, use the "One tail comma alpha ""One tail, α" column with a negative sign. If the hypothesis test is right-tailed, use the "One tail comma alpha ""One tail, α" column with a positive sign. If the hypothesis test is two-tailed, use the "Two tails comma alpha ""Two tails, α" column with a negative and a positive sign.
Use a χ2-test to test the claim σ<43 at the α=0.10 significance level using sample statistics s=43.7 and n=12.Assume the population is normally distributed.
Identify the test statistic.=11.361 critical value(s).= 5.5785.578 Fail to reject H0. There is not enough evidence at the 10% level of significance to support the claim.
Explain how to decide when a normal distribution can be used to approximate a binomial distribution.
If np≥5 and nq≥5, the normal distribution can be used.
Explain the difference between the z-test for μ using rejection region(s) and the z-test for μ using a P-value.
In the z-test using rejection region(s), the test statistic is compared with critical values. The z-test using a P-value compares the P-value with the level of significance α.
You are testing a claim and incorrectly use the normal sampling distribution instead of the t-sampling distribution. Does this make it more or less likely to reject the null hypothesis? Is this result the same no matter whether the test is left-tailed, right-tailed, or two-tailed? Explain your reasoning.
More likely; for degrees of freedom less than30, the tail of the curve are thicker for a t-sampling distribution.Therefore, if you incorrectly use a standard normal samplingdistribution, the area under the curve at the tails will be smaller than what it would be for thet-test, meaning the criticalvalue(s) will lie closer to the mean The result is the same. In each case, the tail thickness affects the location of the critical value(s).
Find the critical value(s) and rejection region(s) for the indicated t-test, level of significance α, and sample size n. Left-tailed test, α=0.10, n=
The critical value(s) is −1.333.
Find the critical value(s) for a left-tailed z-test with α=0.05. Include a graph with your answer
The critical value(s) is(are) -1.65
Find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer. Right-tailed test, α=
The critical value(s) is/are z=1.751.75. (Round to two decimal places as needed. Use a comma to separate answers as needed.) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Round to two decimal places as needed.) The rejection region is z>1.751.75.
A baseball team claims that the mean length of its games is 1.1 hours. State H0 and Ha in words and in symbols. Then determine whether the hypothesis test for this claim is left-tailed, right-tailed, or two-tailed. Explain your reasoning.
The null hypothesis expressed in words is, "the mean length of a baseball team's games is 1.1 hours." The null hypothesis is expressed symbolically as, "H0: μ=1.1."The alternative hypothesis expressed in words is, "the mean length of a baseball team's games is not equal to 1.1 hours." The alternative hypothesis is expressed symbolically as, "Ha: μ≠1.1."
Find the critical value(s) and rejection region(s) for the type of z-test with level of significance α. Include a graph with your answer. Two-tailed test, α=
The rejection regions are z<negative 2.58−2.58 and z>2.582.58.
Use a t-test to test the claim about the population mean μ at the given level of significance α using the given sample statistics. Assume the population is normally distributed. Claim: μ≠25; α=0.05 Sample statistics: x=25.9, s=4.3, n=
The standardized test statistic is . 69.69. Part 3 P-value=0.5030.503 Fail to reject H0. There is not enough evidence to support the claim.
For the statement below, write the claim as a mathematical statement. State the null and alternative hypotheses and identify which represents the claim. A laptop manufacturer claims that the mean life of the battery for a certain model of laptop is less than 6 hours.
p < 6
Use the given statement to represent a claim. Write its complement and state which is H0 and which is Ha. p > 0.66
p ≤ 0.66
What are the two decisions that you can make from performing a hypothesis test?
reject the null hypothesis. fail to reject the null hypothesis
A used car dealer says that the mean price of a three-year-old sports utility vehicle is $23,000. You suspect this claim is incorrect and find that a random sample of 20similar vehicles has a mean price of $23,710 and a standard deviation of $1924.Is there enough evidence to reject the claim at α=0.05? Complete parts (a) through (e) below. Assume the population is normally distributed.
t0=−2.093, 2.093 t<- 2.093−2.093 and t=1.65 Fail to rejectH0 because the test statistic is not in the rejection region(s).At the 5% level of significance, there is not sufficient evidence to reject the claim that the mean price is $23,000.
Determine the rejection region(s). Select the correct choice below and fill in the answer box(es) within your choice.
t<−1.333.
Use the given statement to represent a claim. Write its complement and state which is H0 and which is Ha. μ≤285
μ>285
Use the given statement to represent a claim. Write its complement and state which is H0 and which is Ha. σ = 2
σ ≠ 2
For the statement below, write the claim as a mathematical statement. State the null and alternative hypotheses and identify which represents the claim. The standard deviation of the base price of a certain type of all-terrain vehicle is at least $326.
σ ≥ 326
In hypothesis testing, does choosing between the critical value method or the P-value method affect your conclusion? Explain.
No, because both involve comparing the test statistic's probability with the level of significance.
Find the P-value for a left-tailed hypothesis test with a test statistic of z=−1.86. Decide whether to reject H0 if the level of significance is α=0.05.
P-value=. 0314.0314 State your conclusion. Choose the correct answer below. Since P≤α, reject H0.