One sample test of hypothesis

Calculate the test statistic z for a population mean of 30, a population standard deviation of 6, a sample mean of 27 and a sample size of 20.

-2.24

Find the p-value (to two significant digits) for the following test. H0: μ = 0, H1: μ ≠ 0, σ = 1, z = 2.06 Hint: the population follows the standard normal distribution.

0.04

Match the level of significance to the type of research for which it is traditionally chosen.

0.05 → consumer research 0.01 → quality assurance 0.10 → political polling

Find the p-value (to two significant digits) for the following test. H0: μ ≤ 0, H1: μ > 0, σ = 1, z = 1.5 Hint: the population follows the standard normal distribution.

0.07

Although the p-value does not give us the probability that the null hypothesis is true, it does suggest the likelihood that it is true. Match the p-value to the strength of the case in favor or the null hypothesis.

0.20 → A good chance H0 is correct. 0.05 → A slight chance H0 is true. 0.01 → Very poor chance H0 is true. 0.001→ Extremely poor chance H0 is true.

Hypothesis testing follows a six step procedure. Place these steps in order (first at the top)

1. State null and alternate hypothesis. 2. Select a level of significance. 3. Identify the test statistic. 4. Formulate a decision rule. 5. Take a sample, and use it to decide. 6. Interpret the result

Hypothesis testing follows a six step procedure. Place these steps in order (first at the top).

1. State null and alternate hypothesis. 2. Select a level of significance. 3. Identify the test statistic. 4. Formulate a decision rule. 5. Take a sample, and use it to decide. 6. Interpret the result.

Calculate the test statistic z for a population mean of 5.5, a population standard deviation of 2, a sample mean of 6.7 and a sample size of 10.

1.90

Suppose a hypothesis test, using α = 0.05, is being conducted with the following null hypothesis: H0: μ = 2. Which one of the following confidence intervals would lead to rejecting the null hypothesis?

A 95% confidence interval of 3 to 4.

Which one of the following best describes Hypothesis Testing?

A procedure based on sample evidence and probability to see if a hypothesis is a reasonable statement.

In the context of hypothesis testing, what is a test statistic?

A value, determined from sample information, used to test the null hypothesis.

How does the use of a two-tailed test effect the size of the rejection area?

Because there are two tails and the total significance remains the same, the area in each tail is half as much

A tire manufacturer claims its new tire has an average tread life of 80,000. To test to see if the process is true, the company conducts a hypothesis test using the following hypotheses: Ho: μ=80,000 H1: μ≠80,000 Which of the following is the result of a Type II error?

Concluding the population average is 80,000 when it really isn't.

A soda bottling company fills bottles with 12 ounces of soda. Overfilling causes the company to give away free soda. Underfilling causes the company to cheat the customer. To test to see if the process is working correctly, the company conducts a hypothesis test using the following hypotheses: Ho: μ=12 H1: μ≠12 Which of the following is the result of a Type I error?

Concluding the population average is not 12 when it really is.

Choose the statements that correctly reflect traditional usage of significance level in statistical research. Select all that apply.

Consumer research uses a 5% significance. Quality control uses a 1% significance level.

A brand of chocolate bar has a stated weight of 6 oz. with σ = 0.25 oz. A sample of 9 bars has an average weight of 6.05 oz. Test H0: μ = 6 oz. H1: μ ≠6 oz. at the 5% significance level.

Do not reject the null hypothesis

A bag of potatoes has a stated weight of 10 pounds with σ = 0.75 pound. A sample of 30 bags has an average weight of 10.3 pounds. Conduct a hypothesis test using a 1% significance level for: H0: μ = 10 H1: μ ≠ 10

Fail to reject the null hypothesis

If the test statistic for a hypothesis test (H0: μ = 11) is z = 1.82 and the critical value for 5% significance is zc = 1.96, what should our conclusion be?

Fail to reject the null hypothesis.

The null hypothesis is not rejected if the confidence interval does not include the hypothesized value.

False

What is the correct procedure when you want to do a test of the population mean but the population standard deviation is unknown? Select all that apply.

Find the sample standard deviation and use it as an estimate for the population standard deviation. The t-distribution is used in place of the z-distribution, assuming a normal population.

Suppose you are conducting a test with α=0.10, which of the following are true? Select all that apply.

For H0: μ≥25, the lower tail should include a rejection region equal to 0.10. For H0: μ=25, both the upper and lower tails include a rejection region equal to 0.05.

A cereal manufacturer tests a sample of 50 boxes of cereal to see if the average weight per box is 14 ounces. Which of these would be a valid null or alternative hypothesis for the given scenario?

H0: μ = 14

Which one of the following sets of hypotheses requires the use of a two-tailed test?

H0: μ = 5.6, H1: μ ≠ 5.6

Which of the following sets of hypotheses require the use of a one-tailed test? Select all that apply.

H0: μ ≥ 7.5, H1: μ < 7.5 H0: μ ≤ -25, H1: μ > -25

A one-tailed and a two-tailed test have different critical values at the same significance level. Match the critical values to the to the correct test. z=1.96 or -1.96 z=-2.33 z=2.58 or -2.58 z=1.65

H0:μ = 0, H1:μ ≠ 0, α=0.05 H0:μ=0, H1:μ≠0, α=0.01 H0:μ≥0, H1:μ<0, α=0.01 H0:μ = 0, H1:μ ≠ 0, α=0.05

Which of the following statements of a test hypothesis uses the correct protocol for stating an alternate hypothesis?

H1: μ ≠ 24 H1: μ < 30

Which of the following statements of a test hypothesis violates the protocol for choosing null and alternate hypotheses?

H1: μ ≤ -2

Which of the following statements accurately describe the p-value? Select all that apply.

If we reject the null, it is the probability of making a Type I error. The null is rejected when the p-value is less than α.

What is the last step in the six step Hypothesis testing procedure?

Interpret the result.

Which of the following is the best definition of the null hypothesis?

It is a statement that is not rejected unless the sample data provide convincing evidence that it is false.

The significance level refers to the total area under the distribution in the region of rejection. What happens to this area in a two-tailed test?

It is divided into two equal parts, each half as large, one under each tail.

Which of the following are essential elements of hypothesis testing? Select all that apply.

It makes use of sample data. It uses probability theory to determine if a hypothesis is a reasonable statement.

The result of a hypothesis test was traditionally reported as either "reject H0" or "fail to reject H0". What is the purpose of reporting the p-value in addition?

It provides additional insight into the strength of the decision.

Which of the following statements are valid descriptions of the alternate hypothesis? Select all that apply.

It tells what you will conclude if you reject the null hypothesis. It is written symbolically as H1.

Suppose you are performing a hypothesis test with σ unknown, n=29, α=0.01, and the following hypotheses: H0: μ ≤ 24 H1: μ > 24 What is the decision rule?

Reject H0 if the test statistic is greater than 2.467.

Suppose you are performing a hypothesis test with σ unknown, n=22, α=0.05, and the following hypotheses: H0: μ = 24 H1: μ ≠ 24 What is the decision rule?

Reject H0 if the test statistic is less than -2.080 or greater than 2.080.

Suppose you are performing a hypothesis test with σ unknown, n = 23, α = 0.10, and the following hypotheses: H0: μ ≥ 24 H1: μ < 24 What is the decision rule?

Reject H0 if the test statistic is less than −1.321.

We wish to test H0: μ ≥ 30 and H1: μ < 30 at the 0.05 level of significance. Which of these statements are correct? Select all that apply.

Reject H0 if z < -1.65 Fail to reject H0 if z > -1.65

We wish to test H0: μ ≤ 12 and H1: μ > 12 at the 0.05 level of significance. Which of these statements are correct? Select all that apply.

Reject H0 if z > 1.65 Fail to reject H0 if z < 1.65

A model of car claims mileage of 24 mpg. with σ = 4 mpg. A sample of 4 cars got an average of 20.5 mpg. Test H0: μ = 24 H1: μ ≠ 24 at the 10% significance level.

Reject the null hypothesis

A paint manufacturer claims that a gallon of their paint will cover at least 1200 square feet of smooth wall with σ = 80 square feet. Thirty-six gallons of paint were tested and the average square feet covered was 1175. Conduct a hypothesis test using a 5% significance level for: H0: μ ≥ 1200 H1: μ < 1200

Reject the null hypothesis

If the test statistic for a hypothesis test (H0: μ = 25) is z = 2.05 and the critical value for 5% significance is 1.96, what should our conclusion be?

Reject the null hypothesis and conclude the alternative is true.

Knowing the p-value allows us to asses the weight of evidence against the null hypothesis. Match the p-value to the relative strength of evidence. 0.20 0.05 0.01 0.001

Some evidence against H0 Strong evidence against H0. Very strong evidence against H0. Extremely strong evidence against H0.

What is the "critical value" for a hypothesis test?

The dividing point between rejecting and failing to reject the null hypothesis.

A tire manufacturer claims its new tire has an average tread life of 80,000. To test to see if the process is true, the company conducts a hypothesis test using the following hypotheses: Ho: μ=80,000 H1: μ≠80,000 If the null is rejected, what would the interpretation be?

The manufacturer's claim is not true - the population average is not 80,000

Which of the following is a "hypothesis" in the statistical sense? Select all that apply.

The population mean miles per gallon of a particular car is 44. The population mean lifetime of a particular brand of light bulb is at least 1000 hours.

What is the meaning of "level of significance" in the context of hypothesis testing?

The probability of rejecting the null hypothesis when it is true.

What is the p-value?

The probability that a sample value would be as far or further from the expected value, given that the null hypothesis is true.

Choose the statement that best explains the use of the sample standard deviation in tests of the mean where the population standard deviation is not known.

The sample standard deviation is used as an estimate for the population standard deviation.

What is a "decision rule" in the context of hypothesis testing?

The specific conditions under which the null hypothesis is to be rejected.

From the statements below, select all that are accurate descriptions of the null hypothesis.

The term "null" refers to no significant difference. The null hypothesis is designated H0. The purpose of the test is to prove the null hypothesis is false.

A soda bottling company fills bottles with 12 ounces of soda. Overfilling causes the company to give away free soda. Underfilling causes the company to cheat the customer. To test to see if the process is working correctly, the company conducts a hypothesis test using the following hypotheses: Ho: μ=12 H1: μ≠12 If we fail to reject the null, what would the interpretation be?

There is insufficient evidence to conclude that the mean is different from 12.

Failing to reject a null hypothesis when it is false is referred to as ___.

Type II error

Select all statements that correctly describe the null hypothesis.

We either "reject" or "fail to reject" it, we cannot say that we "accept it" or that it is "true". It is developed for testing purposes.

The formula that is used to find the statistic for a Type II error is z = Xc−μ1 σ/n√ Match the variable to its description.

Xc → the critical value for H0: μ=μ0 μ1→ the mean of the actual dist. σ → population standard deviation n → sample size μ0 → the mean of the assumed dist.

Choose the formula that is used to find the test statistic for a mean when the population standard deviation is unknown.

t = (x−μ/s) n√

Which one of these formulas would you use to calculate the test statistic for a test of the mean with the population standard deviation unknown?

t = X−μ/ s/n√

When testing a mean, where the population standard deviation is known, we calculate the test statistic using the formula z = X−μσ/n√X-μσ/n. Match the variables to their description.

z → The test statistic X → The sample mean μ → The population mean σ/n⎯√n → The standard error

Which symbol represents the level of significance?

α

The probability of rejecting a true null is know as which of the following? Select all that apply.

α Type I error