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Chapter 7: Problem 18

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 \(30 ?\)

Short Answer

Expert verified
When the sample size is less than 30, the sampling distribution of the sample mean will be nearly normal if the data comes from a population with a normal distribution.

Step by step solution

01

Condition Identification

The conditions required for the sampling distribution of the sample mean to be nearly normal when the sample size is less than 30: 1. The data must come from a population with a known (or assumed) normal distribution. If the population distribution is clearly not normal (e.g., it is skewed or bimodal), the sampling distribution of the sample mean may not be normal. 2. The sample size must be sufficiently large. Generally, larger sample sizes lead to sampling distributions that are more closely approximate a normal distribution. However, when the sample size is less than 30, a normal population distribution is critical.

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Most popular questions from this chapter

In a large city, \(88 \%\) of the cases of car burglar alarms that go off are false. Let \(\hat{p}\) be the proportion of false alarms in a random sample of 80 cases of car burglar alarms that go off in this city. Calculate the mean and standard deviation of \(\hat{p}\), and describe the shape of its sampling distribution.

A population of \(N=5000\) has \(\sigma=25\). 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. \(n=300\) b. \(n=100\)

Let \(x\) be a continuous random variable that has a normal distribution with \(\mu=48\) and \(\sigma=8\). Assuming \(n / N \leq .05\), find the probability that the sample mean, \(\bar{x}\), for a random sample of 16 taken from this population will be a. between \(49.6\) and \(52.2\) b. more than \(45.7\)

In a population of 18,700 subjects, \(30 \%\) possess a certain characteristic. In a sample of 250 subjects selected from this population, \(25 \%\) possess the same characteristic. How many subjects in the population and sample, respectively, possess this characteristic?

The amounts of electricity bills for all households in a city have a skewed probability distribution with a mean of \(\$ 140\) and a standard deviation of \(\$ 30\). Find the probability that the mean amount of electric bills for a random sample of 75 households selected from this city will be a. between \(\$ 132\) and \(\$ 136\) b. within \(\$ 6\) of the population mean c. more than the population mean by at least \(\$ 4\)
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