MathStat474 - HW 12

Posted by Apr 10. Due: WED April 17.

Remember to always explain your answers.

Problem 1. [Random sampling conditions for hypothesis testing.]

We plan to poll 200 students enrolled in statistics at your college by distributing surveys during class. Which of the following hypotheses could be tested with the survey results? Mark each as valid (OK to use to test the hypothesis) or not valid (should not be used to test the hypothesis). Of course, explain why valid/invalid!

  • The hypothesis that p = 0.60, where p is the proportion of students at your college who study the recommended “2 hours a week for each class unit.”

  • The hypothesis that p = 0.60, where p is the proportion of students at your college who visit a dentist at least once a year.

  • The hypothesis that p = 0.60, where p is the proportion of students at your college who are receiving some form of financial aid.

  • The hypothesis that p = 0.60, where p is the proportion of students at your college who spend more than $600 per semester on textbooks.

Problem 2. [conditions for using the z-test for population proportion]

In each of the following scenarios, you need to decide whether it is appropriate to use the z-test for the population proportion p, and if not, which condition is violated.

Scenario 1: The UCLA Internet Report (February 2003) estimated that roughly 8.7% of Internet users are extremely concerned about credit card fraud when buying online. Has that figure changed since? To test this, a random sample of 100 Internet users was chosen. When interviewed, 10 said that they were extremely worried about credit card fraud when buying online. Let p be the proportion of all Internet users who are concerned about credit card fraud.

Which one of the following statements is correct about using the z-test for p?

  • It is safe to use the z-test for p.
  • It is not safe to use the z-test for p, since the sample is not a random sample from the entire population (or cannot be considered as one).
  • It is not safe to use the z-test for p since \(n p_0\) is not large enough.
  • It is not safe to use the z-test for p since \(n(1 - p_0)\) is not large enough.

Scenario 2: The UCLA Internet Report (February 2003) estimated that a proportion of roughly .75 of Internet-using homes are still using dial-up access, but claimed that the use of dial-up is declining. Is that really the case? To examine this, a follow-up study was conducted a year later in which out of a random sample of 1,308 households that had an Internet connection, 804 were connecting using a dial-up modem. Let p be the proportion of all U.S. Internet-using households that have dial-up access.

Which one of the following statements is correct about using the z-test for p?

  • It is safe to use the z-test for p.
  • It is not safe to use the z-test for p, since the sample is not a random sample from the entire population (or cannot be considered as one).
  • It is not safe to use the z-test for p since \(n p_0\) is not large enough.
  • It is not safe to use the z-test for p since \(n(1 - p_0)\) is not large enough.

Problem 3. [p-value and complete testing setup - 10 points for this problem]

We would like to find out the following: Has the proportion of U.S. adults who support the death penalty for convicted murderers changed since 2003, when it was 0.64? We take a random sample of 1000 US adults and learn that 675 are in favor of the death penalty for convicted murderers.

  • What is the population parameter we are testing?
  • Set up the null and alternative hypothesis.
  • State the test statistic (formula) and compute the observed value of the test statistic.
  • Compute the \(p\)-value of this data.
  • Draw your conclusion.

Problem 4. [this is about conficendce intervals and margin of error]

  • What sample size of U.S. adults do you need, if you would like to estimate the proportion of U.S. adults who are “pro-choice” with a 2.5% margin of error (at the 95% level)?
  • Your answer to the above question indicates that if you take a sample of that size, the sample proportion of adults who are pro-choice is (select one):
    • more than 2.5% away from the proportion who are pro-choice among all U.S. adults.
    • within 2.5% of the proportion who are pro-choice among all U.S. adults.
    • exactly equal to the proportion who are pro-choice among all U.S. adults.
  • If you were to use a random sample of size n = 640 U.S. adults (instead of what you found in question 1), what would the margin of error roughly be?
    • 4%
    • .156%
    • 3%
    • 5%

Typing up work

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