According to a U.S. news poll, 38% of students in the class of 2013 had done an internship during their time as an undergraduate student. Dana is interested in finding out whether students at her university had an internship rate that was higher than the national average. She obtained a list of 25 randomly selected students from the population of all students at her university by requesting this information from the university’s institutional research office. She collected the responses and calculated that the proportion for her university was 43%.Which one of the following statements about the z-test is correct? It is not safe to use the z-test for p, since n * po is not large enough. 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 * (1 − po) is not large enough.
Question
According to a U.S. news poll, 38% of students in the class of 2013 had done an internship during their time as an undergraduate student. Dana is interested in finding out whether students at her university had an internship rate that was higher than the national average. She obtained a list of 25 randomly selected students from the population of all students at her university by requesting this information from the university’s institutional research office. She collected the responses and calculated that the proportion for her university was 43%.Which one of the following statements about the z-test is correct? It is not safe to use the z-test for p, since n * po is not large enough. 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 * (1 − po) is not large enough.
Solution
The z-test is used to determine whether two population means are different when the variances are known and the sample size is large. The test statistic is assumed to have a normal distribution, and nuisance parameters such as standard deviation should be known in order for an accurate z-test to be performed.
In this case, the sample size (n) is 25 and the proportion of students who had an internship (p) is 43% or 0.43. The national average proportion (p0) is 38% or 0.38.
The conditions for a z-test are:
- The sampling method is simple random sampling.
- The samples are independent.
- Each sample includes at least 10 successes and 10 failures.
- The population is at least 20 times as big as the sample.
In this case, the sample is a simple random sample, and the samples are independent. The sample includes more than 10 successes (students who had an internship) and 10 failures (students who did not have an internship). The population is presumably at least 20 times as big as the sample.
The conditions for the z-test are met, so it is safe to use the z-test for p.
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