13. The programmer at a local public radio station is compiling data on listeners who stream podcasts. An executive for the national broadcaster has communicated that the overall population mean is 20,500 with a standard deviation of 2,180. The local programmer has a sample of 30 podcasts for her station. By the central limit theorem, which interval do about 99.7% of the sample means fall within? A. 20,102 and 20,898
B. 20,282 and 20,718
C. 19,306 and 21,694
D. 19,704 and 21,296

C. 19,306 and 21,694

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question, for samples of 30:

Mean: 20,500

Standard deviation:

99.7% of the sample means fall within 3 standard deviations of the mean, by the Empircal Rule with the Central Limit Theorem:

Lower bound: 20500 - 3*398 = 19,306

Upper bound: 20500 + 3*398 = 21,694

So the correct answer is option C.

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step-by-step explanation: