Question

Suppose we sample 16 values from a normal distribution with a

mean of 100 and a standard deviation of 20. What is the probability
that we see a sample mean less than 95? Round your answer to 4
decimal

Answer 1

To find the probability of a sample mean being less than 95 from a normally distributed population with a mean of 100 and a standard deviation of 20, calculate the standard error, find the Z-score, and then look up the probability. The calculated probability is 0.1587.

Step-by-step explanation:

The question pertains to the concept of the sampling distribution of the sample mean. Since the population distribution is normal, the sample means will also be distributed normally (Central Limit Theorem). To find the probability that the sample mean is less than 95, we can use the Z-score formula for sample means:

1. Calculate the standard error of the mean by dividing the population standard deviation by the square root of the sample size (In this case, Standard Error = 20/sqrt(16) = 5).
2. Calculate the Z-score using the formula Z = (Sample Mean - Population Mean) / Standard Error (Z = (95 - 100) / 5 = -1).
3. Look up the Z-score in a standard normal distribution table or use a statistical software to find the probability corresponding to Z = -1, which is approximately 0.1587.

The probability that the sample mean is less than 95 would be 0.1587, or rounded to four decimal places, 0.1587.