Chapter 9: Section 9.1 Hypothesis Testing

H<sub>0

Uses an equal sign</sub>

Null Hypothesis

H<sub>a

Uses not equal signs. They can be greater, less or not equal to</sub>

Alternative Hypothesis

Used when < or > is used for the alternative hypothesis

One sided

Used when ≠ is used for the alternative hypothesis

Same result as creating a confidence interval!

Two sided test

Multiple Choice

Which of the following shows a right tailed test?

• H₀: μ < 15

• H₀: μ > 15

• H_a: μ < 15

• H_a: μ > 15

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The US Bureau of Labor Statistics estimates that 69.7% of high school graduates enroll in a college or university. Bernard has great pride in the quality of his large high school. He thinks the proportion of college-bound students is greater than last year's graduating class.

State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

Multiple Select

Select which of the following are correct ways to write null and altnerative hypotheses:

Multiple Choice

In a hypothesis test, the decision between a one-sided and a two-sided alternative hypothesis is based on:

• Which one gives you a significant result

• the alternative hypothesis appropriate for the context of the problem

• how accurate you wish the results of the test to be

• the level of significance of the test

Assuming that the null hypothesis is true, what is the probability of getting a sample that matches the sample result (which should match your alternative hypothesis)

​

What is a p-value?

page 4

We are performing a hypothesis test with the following Ho and Ha.
where p = the true proportion of all last year's graduates at Bernard's school who enroll in a college or university. Bernard asks a random sample of 40 of last year's graudates from his high school if they are enrolled in a college or university and 34 say "yes". The sample proportion enrolled in a college or university is p hat = 34/40 = 0.85.

(a) Explain what it would mean for the null hypothesis to be true in this setting.

page 4

We are performing a hypothesis test with the following Ho and Ha.
where p = the true proportion of all last year's graduates at Bernard's school who enroll in a college or university. Bernard asks a random sample of 40 of last year's graudates from his high school if they are enrolled in a college or university and 34 say "yes". The sample proportion enrolled in a college or university is p hat = 34/40 = 0.85.

Bernard performed a significance test and obtained a P-value of 0.018.

(a) Interpret the P-value.

Multiple Choice

In a sample with the given H0 and Ha, the following p hat was calculated. The resulting P-value is 0.18. What is the correct interpretation of this P-value?

• Only 18% of the city residents support the tax increase.

• There is an 18% chance that the majority of residents supports the tax increase.

• Assuming that 50% of residents support the tax increase, there is an 18% probability that the sample proportion would be 0.527 or greater by chance alone.

• Assuming that more than 50% of residents support the tax increase, there is an 18% probability that the sample proportion would be 0.527 or greater by chance alone.

• Assuming that 50% of residents support the tax increase, there is an 18% chance that the null hypothesis is true by chance alone.

Open Ended

Mr. Tabor believes that less than 75% of the students at his school completed their math homework last night.

The math teachers inspect the homework assignments from a random sample of 50 students at the school. Only 68% of the students completed their math homework. A significance test yields a P-value of 0.1265.

Interpret the P-value.

Compare P-value to significance level.

If P-value < α --reject NULL (evidence has been found)

If P-value > ​α -- fail to reject NULL (not enough evidence has been found)

Making conclusions

page 5

The police department in a certain city is trying to determine if it is worth the cost to install a speed sensor and a traffic camera on a highway near the city. They will install the speed sensor and traffic camera is more than 20% of cars are speeding. The police department selects a random sample of 100 cars on the highway, measures their speed, and finds that 28 of the 100 cars are speeding. A significance test is performed using the hypotheses given, where p is the true proportion of all cars on this highway that are speeding.
The resulting P-value is 0.023. What conclusion would you make at the 0.05 significance level?

Open Ended

A Gallup poll report revealed that 72% of teens said they seldom or never argue with their friends. Yvonne wonders whether this result holds true in her large high school, so she surveys a random sample of 150 students at her school. Yvonne finds that 96 of the 150 students (64%) say they rarely or never argue with friends. A significance test yields a P-value of 0.0291.

What conclusion would you make at the α = 0.10 level?

Multiple Select

A significance test yields a P-value of 0.0589.

Select all the following that are true.

• Reject the null at α = 0.1

• Fail to reject the null at α = 0.1

• Reject the null at α = 0.05

• Fail to reject the null at α = 0.05

Multiple Choice

If the P-value of a test of significance is greater than the level of significance, then which of the following conclusions is appropriate?

I. Fail to reject the null hypothesis

II. Accept the null hypothesis

III. The null hypothesis is true

• II only

• III only

• II and III

• I only

• I, II and III

Multiple Choice

When p-value is greater than alpha we:

• Reject H₀

• Fail to reject H₀

• Reject H_a

• Fail to reject H_a

Multiple Choice

In a research report, the term "significant" is used when the null hypothesis is rejected.

• True

• False

​Type I and II Errors:
Remember this chart!

page 5

The police department in a certain city is trying to determine if it is worth the cost to install a speed sensor and a traffic camera on a highway near the city. They will install the speed sensor and traffic camera is more than 20% of cars are speeding. The police will select a random sample of 100 cars on the highway and perform a test at the 0.05 significance level where p is the true proportion of all cards on this highway that are speeding. Describe a Type I error and a consequence.

page 5

The police department in a certain city is trying to determine if it is worth the cost to install a speed sensor and a traffic camera on a highway near the city. They will install the speed sensor and traffic camera is more than 20% of cars are speeding. The police will select a random sample of 100 cars on the highway and perform a test at the 0.05 significance level where p is the true proportion of all cards on this highway that are speeding. Describe a Type II error and consequence.

Open Ended

You are thinking about opening a restaurant and are searching for a good location. From research you have done, you know that the mean income of those living near the restaurant must be over $85,000 to support the type of upscale restaurant you wish to open. You decide to take a simple random sample of 50 people living near one potential location. Based on the mean income of this sample, you will perform a hypothesis test with given null and alternative hypotheses.

where μ is the true mean income in the population of people who live near the restaurant. Describe a Type II error in this setting and a possible consequence.

Open Ended

You are thinking about opening a restaurant and are searching for a good location. From research you have done, you know that the mean income of those living near the restaurant must be over $85,000 to support the type of upscale restaurant you wish to open. You decide to take a simple random sample of 50 people living near one potential location. Based on the mean income of this sample, you will perform a hypothesis test with given null and alternative hypotheses.

where μ is the true mean income in the population of people who live near the restaurant. Describe a Type I error in this setting and a possible consequence.

Multiple Choice

If no alpha level is given, what alpha level is assumed?

• 0.01

• 0.25

• 0.05

• 0.10

Multiple Choice

Define a Type I error.

• The failure to reject a false null hypothesis

• Rejecting a true null hypothesis

• The acceptance to reject a null hypothesis

• Accepting a true null hypothesis

Multiple Choice

A test is made of H₀ : µ = 50 versus H₁ : µ ≠ 50 is performed using α = 0.01 significance level. The value of the test statistic is z = -2.68 and it is concluded that we fail to reject H₀ .

If the true value of µ is 65, is the conclusion a Type I Error, Type II error, or a correct decision?

• Type I

• Type II

• Correct decision

Multiple Choice

A test is made of H₀ : µ = 17 versus H₁ : µ < 17 is performed using α = 0.01 significance level. The value of the test statistic is z = -2.68 and it is concluded that H₀ is rejected.

If the true value of µ is 10, is the conclusion a Type I Error, Type II error, or a correct decision?

• Type I

• Type II

• Correct decision

Multiple Choice

A test is made of H₀ : µ = 17 versus H₁ : µ < 17 is performed using α = 0.01 significance level. The value of the test statistic is z = -2.68 and it is concluded that H₀ is rejected.

If the true value of µ is 17, is the conclusion a Type I Error, Type II error, or a correct decision?

• Type I

• Type II

• Correct Decision