A study suggests that 25% of 25-year-olds have gotten married. You believe that this is incorrect and decide to collect your own sample for a hypothesis test. From a random sample of 25-year-olds in census data with a size of 776, you find that 24% of them are married. A friend of yours offers to help you with setting up the hypothesis test and comes up with the following hypotheses. Indicate any errors you see. (See below.)
True or False: The null hypothesis should use p and not p-hat since we are interested in the population proportion, not the sample proportion.
True or False: The null value should be what we are testing (0.25), not the observed value (0.24).
400 students were randomly sampled from a large university, and 289 said they did not get enough sleep. Conduct a hypothesis test to check whether this represents a statistically significant difference from 50% and use a significance level of 0.01.
True or False: The correct hypotheses are HO: μ = 0.5, HA: μ ≠ 0.5.
True or False: α = 0.01.
True or False: Simple random sample gets us independence, and the success-failure conditions are satisfied since 289 and 400-289 = 111 are both at least 10.
What is p-hat? 0.7225.
What is SE? 0.025.
What is Z? 8.9.
True or False: The p-value is nearly zero.
True or False: Because the p-value is less than α = 0.01, we reject the null hypothesis and conclude that the fraction of students who don't get enough sleep is different than 50%. Because the observed value is greater than 50% and we have rejected the null hypothesis, we can conclude that a majority of students at the surveyed university do not get enough sleep.
In each part below, there is a value of interest and two scenarios (I and II). For each part, report if the value of interest is larger under scenario I, scenario II, or whether the value is equal under the scenarios.
The standard error of p-hat when (I) n = 125 or (II) n = 500.
Scenario (I) is larger.
Scenario (II) is larger.
The value is equal under both scenarios.
The margin of error of a confidence interval when the confidence level is (I) 90% or (II) 80%.
Scenario (I) is larger.
Scenario (II) is larger.
The value is equal under both scenarios.
The p-value for a Z-statistic of 2.5 calculated based on a (I) sample with n = 500 or based on a (II) sample with n = 1000.
Scenario (I) is larger.
Scenario (II) is larger.
The value is equal under both scenarios.
The probability of making a Type 2 Error when the alternative hypothesis is true, and the significance level is (I) 0.05 or (II) 0.10.
Scenario (I) is larger.
Scenario (II) is larger.
The value is equal under both scenarios.