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In 2017, 20.1% of all high school students in the U.S. had a job. A school official from a district thought that the percentage of high school students who had a job in that school district was higher than the national proportion. She took a random sample of 285 high school students and found that 64 of had a job.Is this survey valid or not valid for testing the hypothesis that the proportion of the high school students who worked in that district is higher than the national average?

Question

In 2017, 20.1% of all high school students in the U.S. had a job. A school official from a district thought that the percentage of high school students who had a job in that school district was higher than the national proportion. She took a random sample of 285 high school students and found that 64 of had a job.Is this survey valid or not valid for testing the hypothesis that the proportion of the high school students who worked in that district is higher than the national average?

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Solution

To determine if the survey is valid for testing the hypothesis, we need to check if it meets the conditions for a hypothesis test for a proportion. These conditions are:

  1. Random: The sample should be randomly selected. In this case, the problem states that the school official took a random sample, so this condition is met.

  2. Normal: The sampling distribution should be approximately normal. This condition is met if both np and n(1-p) are greater than or equal to 10, where n is the sample size and p is the assumed proportion in the null hypothesis. Here, n=285 and p=0.201 (20.1% expressed as a decimal). So, np=2850.201=57.285 and n(1-p)=2850.799=227.715. Both of these values are greater than 10, so this condition is met.

  3. Independent: The sample size should be less than 10% of the population size to use the independence assumption. As the population size (all high school students in the U.S.) is much larger than 285, this condition is also met.

Since all the conditions are met, the survey is valid for testing the hypothesis that the proportion of the high school students who worked in that district is higher than the national average.

This problem has been solved

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