91Ó°ÊÓ

A population of \(N=100,000\) has \(\sigma=40\). In each of the following cases, which formula will you use to calculate \(\sigma_{\bar{x}}\) and why? Using the appropriate formula, calculate \(\sigma_{\bar{x}}\) for each of these cases. a. \(n=2500\) b. \(n=7000\)

What condition or conditions must hold true for the sampling distribution of the sample mean to be normal when the sample size is less than \(30 ?\)

A population has a distribution that is skewed to the right. A sample of size \(n\) is selected from this population. Describe the shape of the sampling distribution of the sample mean for each of the following cases. a. \(n=25\) b. \(n=80\) c. \(n=29\)

The amounts of electricity bills for all households in a particular city have an approximately normal distribution with a mean of \(\$ 140\) and a standard deviation of \(\$ 30 .\) Let \(\bar{x}\) be the mean amount of electricity bills for a random sample of 25 households selected from this city. Find the mean and standard deviation of \(\bar{x}\), and comment on the shape of its sampling distribution.

Let \(x\) be a continuous random variable that has a distribution skewed to the right with \(\mu=60\) and \(\sigma=10\). Assuming \(n / N \leq .05\), find the probability that the sample mean, \(\bar{x}\), for a random sample of 40 taken from this population will be a. less than \(62.20\) b. between \(61.4\) and \(64.2\)

The times that college students spend studying per week have a distribution that is skewed to the right with a mean of \(8.4\) hours and a standard deviation of \(2.7\) hours. Find the probability that the mean time spent studying per week for a random sample of 45 students would be a. between 8 and 9 hours b. less than 8 hours

As mentioned in Exercise \(7.33\), among college students who hold part-time jobs during the school year, the distribution of the time spent working per week is approximately normally distributed with a mean of \(20.20\) hours and a standard deviation of \(2.60\) hours. Find the probability that the average time spent working per week for 18 randomly selected college students who hold part-time jobs during the school year is a. not within 1 hour of the population mean b. 20 to \(20.50\) hours c. at least 22 hours d. no more than 21 hours

Let \(\hat{p}\) be the proportion of elements in a sample that possess a characteristic. a. What is the mean of \(\hat{p}\) ? b. What is the formula to calculate the standard deviation of \(\hat{p} ?\) Assume \(n / N \leq .05\). c. What condition(s) must hold true for the sampling distribution of \(\hat{p}\) to be approximately normal?

For a population, \(N=2800\) and \(p=.29 .\) A random sample of 80 elements selected from this population gave \(\hat{p}=.33\). Find the sampling error.

According to the central limit theorem, the sampling distribution of \(\hat{p}\) is approximately normal when the sample is large. What is considered a large sample in the case of the proportion? Briefly explain.