Distributions Identify the distribution (normal, Student t, ch...

Question

Distributions Identify the distribution (normal, Student t, chi-square) that should be used in each of the following situations. If none of the three distributions can be used, what other method could be used?

a. In constructing a confidence interval of , you have 75 sample values and they appear to be from a population with a skewed distribution. The population standard deviation is not known.

Accepted Answer

All right, hello, everyone. So this question says, a sociologist is constructing a confidence interval for the average number of hours worked per week, using a sample of 70 individuals from a population with a less skewed distribution. The population standard deviation is not provided. Which distribution should be used? Option A says chi square distribution, B says student T distribution, C says normal distribution. And D says Poisson distribution. So to understand the answer to a question like this, the first thing we have to do is understand the central limit theorem. Now, what the central limit theorem says is that the sampling distribution of a sample will be approximately normal regardless of the shape of the distribution itself, provided that the sample size is large enough. Now, according to the central limit theorem. The sample size is considered large enough when it is greater than or equal to 30. In this case, our sample is 70, which is of course greater than 30, which means that we can apply the central limit theorem. So here, the sampling distribution is going to be approximately normal, but recall that we can't actually use a normal distribution in this context, because in order for us to be able to use a normal distribution, the standard deviation of the population should be given to us, and in this case it's not. So since there's no population standard deviation, we cannot use a normal distribution. Instead, recall that the student T distribution is appropriate when there is no population standard deviation and when the sample size is large, which in this case it is. So therefore, the student T distribution is most appropriate for this case, which means that option B is the correct answer. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

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Key Concepts

Confidence Interval

Student's t-distribution

Skewed Distribution

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