A population has a mean of 80 and a standard deviation of 7. A sample of 49 observations will be taken. The probability that the sample mean will be larger than 82 is
A)0.5228
B)0.9772
C)0.4772
D)0.0228

Question

Quizplus · Accepted Answer

We need to use the Central Limit Theorem (CLT) to solve this problem. According to CLT, the sampling distribution of the sample means is approximately normal with mean μ and standard deviation σ/√n, where μ is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, we have μ = 80, σ = 7, and n = 49. Therefore, the sampling distribution of the sample means is approximately normal with mean 80 and standard deviation 7/√49 = 1.

Now, we want to find the probability that the sample mean will be larger than 82. We can standardize the sample mean by using the z-score formula:

z = (x̄ - μ) / (σ/√n)

where x̄ is the sample mean. Plugging in the values, we get:

z = (82 - 80) / (1) = 2

We want to find the probability that z is greater than 2. This is equivalent to finding the area to the right of z = 2 under the standard normal distribution curve.

Using a standard normal distribution table or a calculator, we find that the probability is 0.0228 (rounded to four decimal places).

Therefore, the answer is D) 0.0228.

A Population Has a Mean of 80 and a Standard