Question

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?

Answer 1

The interval in which 99.7% of the sample means fall within 3 standard deviations of the mean is 19,306 to 21,694.

What is the central limit theorem?

According to the central limit theorem,

Sample mean = Population mean = μ

Sample standard deviation = (Population standard deviation)/√n = σ/√n

where n is at least 30.

Calculation:

It is given that,

The overall population mean = 20,500

Population standard deviation σ = 2180

Sample size n = 30

Since n = 30, the central limit theorem can be applied.

According to the Empirical rule(68-95-99), 99.7% of the sample means fall within the 3 standard deviations of the mean.

So, the interval is calculated by

Sample mean = Population mean = μ = 20500

Standard deviation(sample) = σ/√n

⇒ S = 2180/√30 =398.011 ≅ 398

Thus,

The lower bound  = 20500 - 3 × 398 = 19,306

The upper bound = 20500 + 3 × 398 = 21,694

Therefore, the required interval is from 19,306 to 21,694.

Learn more about the central limit theorem here:

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