A specific study found that the average number of doctor visits per year for people over 55 is 8 with a standard deviation of 2. Assume that the variable is normally distributed.

1. Identify the population mean.

2. Identify the population standard deviation.

3. Suppose a random sample of 15 people over 55 is selected. What is the probability that the sample mean is above 9?

4. Suppose a random sample of 100 people over 55 is selected. What is the probability that the sample mean will be below 7?

With your data, these are the best estimates.

1. 8

2. 2

3. Z = (score-mean)/SEm

SEm = SD/√n

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability relate to the Z score.

4. Same process as 3.

1. The population mean is the average number of doctor visits per year for people over 55, which is given as 8 in the specific study.

2. The population standard deviation is the measure of variation in the number of doctor visits per year for people over 55 in the population. In this case, it is given as 2 in the specific study.

3. To find the probability that the sample mean is above 9 for a random sample of 15 people, we can use the sampling distribution of the sample mean. The sampling distribution of the sample mean for a large enough sample (n > 30) will be approximately normally distributed.

First, we need to calculate the standard error of the sample mean (SE), which is the population standard deviation (2) divided by the square root of the sample size (√15).

SE = 2 / √15 ≈ 0.5164

Next, we can standardize the sample mean of 9 using the formula: z = (sample mean - population mean) / SE.

z = (9 - 8) / 0.5164 ≈ 1.937

Using a standard normal distribution table or a statistical calculator, we can find the probability associated with the z-score of 1.937. The probability of obtaining a sample mean above 9 is the area under the standard normal distribution curve to the right of 1.937.

P(sample mean > 9) ≈ 1 - P(Z ≤ 1.937)

By looking up the value in the standard normal distribution table or using a statistical calculator, we can find that P(Z ≤ 1.937) is approximately 0.9734. Therefore,

P(sample mean > 9) ≈ 1 - 0.9734 ≈ 0.0266
The probability that the sample mean is above 9 is approximately 0.0266.

4. Similarly, to find the probability that the sample mean will be below 7 for a random sample of 100 people, we can use the same steps as above.

First, calculate the standard error of the sample mean (SE) using the population standard deviation (2) divided by the square root of the sample size (√100).

SE = 2 / √100 = 0.2

Next, standardize the sample mean of 7 using the formula: z = (sample mean - population mean) / SE.

z = (7 - 8) / 0.2 = -5

We want the probability that the sample mean will be below 7, which is the area under the standard normal distribution curve to the left of -5.

P(sample mean < 7) = P(Z ≤ -5)

Using a standard normal distribution table or a statistical calculator, we can find that P(Z ≤ -5) is a very small value close to 0. Therefore,

P(sample mean < 7) ≈ 0

The probability that the sample mean will be below 7 is nearly 0.