1 Sample Z Test for Proportions

Lumber companies dry freshly cut wood in kilns before selling it. A certain percentage of boards develop cracks on their ends during drying. The current drying procedure is known to produce cracks in 16% of boards. The drying supervisor wants to try a new method that she believes will result in a proportion of cracked boards that is less than 16%.

Write the appropriate hypotheses.

• Ho: p = 0.16

  Ha: p = 0.16

• Ho: p = 0.16

  Ha: p =/= 0.16

• Ho: p = 0.16

  Ha: p > 0.16

• Ho: p = 0.16

  Ha: p < 0.16

Simon reads a newspaper report claiming that 12% of all adults in the US are left-handed. He wonders if this figure holds true at his college. Simon chooses an SRS of 100 students and 16 of them are left-handed. He performs a test of Ho: p = 0.12 versus Ha: =/= 0.12. 

The test yields a p-value of 0.2184. What conclusion would you make for a significance level of $\alpha$ = 0.05?

• Since the p-value of 0.2184 > $\alpha$ , we reject the null hypothesis. We have convincing evidence that the proportion of lefties at his college is different from 12%. 

• Since the p-value of 0.2184 > $\alpha$ , we fail to reject the null hypothesis. We do not have convincing evidence that the proportion of lefties at his college is different from 12%. 

• Since the p-value of 0.2184 < $\alpha$ , we reject the null hypothesis. We have convincing evidence that the proportion of lefties at his college is different from 12%. 

• Since the p-value of 0.2184 < $\alpha$ , we fail to reject the null hypothesis. We do not have convincing evidence that the proportion of lefties at his college is different from 12%. 

A Gallup poll report revealed that 72% of teens said they rarely or never argue with their friends. Yvonne wonders if this result is true for her school. She surveys a random sample of 150 students and 64% of them say they rarely or never argue with friends. She uses the data to perform a test of Ho: p = 0.72 versus Ha: p=/= 0.72.

The test yields a p-value of 0.0291. What conclusion would you make for a significance level of 5%?

• Since our p-value of 0.0291 is more than 5%. we fail to reject the null hypothesis. We do not have convincing evidence that the proportion of teens in her school who rarely or never argue with their friends is different from 0.72.

• Since our p-value of 0.0291 is less than 5%. we fail to reject the null hypothesis. We do not have convincing evidence that the proportion of teens in her school who rarely or never argue with their friends is different from 0.72.

• Since our p-value of 0.0291 is more than 5%. we reject the null hypothesis. We have convincing evidence that the proportion of teens in her school who rarely or never argue with their friends is different from 0.72.

• Since our p-value of 0.0291 is less than 5%. we reject the null hypothesis. We have convincing evidence that the proportion of teens in her school who rarely or never argue with their friends is different from 0.72.

In a random of 850 city high school students, 768 said they had access to the internet during school hours. 308 of 355 rural high school students said they had access. which of the following is the p-value for a significance test to determine if these data provide evidence that the proportion of high school students in city areas who have internet access is different than those who go to rural schools?

• 0.002

• 0.011

• 0.022

• 0.033

• 0.066

A nutritionist claims U.S. children drink an average of less than 2 glasses of milk per day. Based on an SRS of 30 children, the p-value of H₀: μ = 2 vs. H_A: μ < 2 is 0.0015. What is the correct interpretation of the p-value?

• The probability that she fails to reject the null hypothesis is 0.015%

• The probability that she rejects the null is 0.015%

• She can be 99.85% confident that the alternative hypothesis is true

• About 0.15% of all samples would product a test statistics at least as extreme as ours if the null hypothesis is true.

• About 99.85% of all samples would product a test statistics at least as extreme as ours if the null hypothesis is true.

A significance test was performed to test the null hypothesis H₀ : p = 0.5 versus the alternative H_a: p > 0.5. The test statistic is z = 1.40. Which of the following is closest to the P-value for this test?

• 0.0808

• 0.1492

• 0.1616

• 0.2984

• 0.9192

Some people say that more babies are born in September than in any other month. To test this claim, you take a simple random sample of 150 students at your school and find that 21 of them were born in September. You are interested in whether the proportion born in September is higher than 1/12—what you would expect if September was no different from any other month. Thus your null hypothesis is H₀: p= 1/12. The P-value for your test is 0.0056. Which of the following

statements best describes what the P-value measures?

• The probability that September birthdays are no more common that any other month is 0.0056.

• The probability that September birthdays are more common is 0.0056

• The probability that the proportion of September birthdays in the population is not equal to 1/12 is 0.0056

• 0.0056 is the probability of getting a sample with a proportion of September birthdays this far or farther above 1/12 if the true proportion is 1/12

• 0.0056 is the probability of getting a sample with a proportion of September birthdays this close to 1/12 if the true proportion is not 1/12

A large nationwide poll recently showed an unemployment rate of 9% in the US. The mayor of a local town wonders if this is true for her town, so she plans to take a sample of residents to see if the unemployment rate is significantly different than 9% in her town. Here are the hypotheses:

 $H_0:\ p\ =\ 0.09$   $H_a:\ p\ \ne0.09$  

Which of the following would be a type I error?

• She concludes the town's unemployment rate is not 9% when it actually is.

• She concludes the town's unemployment rate is not 9% when it actually is not.
  

• She concludes the town's unemployment rate is 9% when it actually is.
  

• She concludes the town's unemployment rate is 9% when it actually is not.
  

A large nationwide poll recently showed an unemployment rate of 9% in the US. The mayor of a local town wonders if this is true for her town, so she plans to take a sample of residents to see if the unemployment rate is significantly different than 9% in her town. Here are the hypotheses:

 $H_0:\ p\ =\ 0.09$   $H_a:\ p\ \ne0.09$  

Which of the following would be a type II error?

• She concludes the town's unemployment rate is not 9% when it actually is.

• She concludes the town's unemployment rate is not 9% when it actually is not.
  

• She concludes the town's unemployment rate is 9% when it actually is.
  

• She concludes the town's unemployment rate is 9% when it actually is not.
  

A large university is curious if they should build another cafeteria. They plan to survey a sample of their students to see if there is strong evidence that the proportion interested is higher than 40%, in which case they would consider building the new cafeteria. Here are the hypotheses they'll use:

 $H_0:\ p\ =\ 0.40$   $H_a:\ p\ >0.40$  

Which of the following would be the consequence of a Type I error?

• They don't consider building the new cafeteria when they should.

• They don't consider building a new cafeteria when they shouldn't.
  

• They consider building a new cafeteria when they shouldn't.
  

• They consider building a new cafteria when they should.