Question

In a survey of citizen satisfaction with local law enforcement, the average score is 65 on a 100-point scale with a population standard deviation (σ) of 8 points. Question: What is the probability that a random sample of 40 citizens will have an average satisfaction score between 63 and 67? Use three decimal places.

Answer 1

To find the probability that a random sample of 40 citizens will have an average satisfaction score between 63 and 67, we calculate the z-scores for both values and find the area under the normal distribution curve between those z-scores. The probability is approximately 0.877.

Step-by-step explanation:

To find the probability that a random sample of 40 citizens will have an average satisfaction score between 63 and 67, we need to calculate the z-scores for both values, and then find the area under the normal distribution curve between those z-scores.

First, we calculate the z-score for 63 using the formula: z = (x - μ) / (σ / √n). Plugging in the values, we get z = (63 - 65) / (8 / √40) = -1.581.

Next, we calculate the z-score for 67 using the same formula: z = (67 - 65) / (8 / √40) = 1.581.

Now, we use a z-table or a calculator to find the area under the normal curve between -1.581 and 1.581. The probability is the area under the curve, and we find that it is approximately 0.877.