Question

CNNBC recently reported that the mean annual cost of auto insurance is 958 dollars. Assume the standard deviation is 144 dollars. You will use a simple random sample of 99 auto insurance policies.

Find the probability that a single randomly selected policy has a mean value between 934.8 and 965.2 dollars.
P(934.8 < X < 965.2) =

Find the probability that a random sample of size n=99n=99 has a mean value between 934.8 and 965.2 dollars.
P(934.8 < ¯xx¯ < 965.2) =

Enter your answers as numbers accurate to 4 decimal places.

Answer 1

Here mean ( )= 958

Standard deviation() = 144

sample size(n) = 99

here to find the probabilities first we need to find the Z - value

by using the Z- value get the probabilities from standard normal tables(Z- tables)

Answer 2

To find the probabilities, we can use the z-score formula and the standard normal distribution table (Z-table).

Given: Mean (μ) = $958 Standard deviation (σ) = $144 Sample size (n) = 99

1. Probability for a single randomly selected policy (X):

We need to find the z-scores for the values 934.8 and 965.2 and then use the Z-table to find the probabilities.

Z-score formula: Z = (X - μ) / (σ / √n)

For X = 934.8: Z1 = (934.8 - 958) / (144 / √99) ≈ -1.4497

For X = 965.2: Z2 = (965.2 - 958) / (144 / √99) ≈ 0.4573

Now, we find the probabilities using the Z-table: P(934.8 < X < 965.2) = P(-1.4497 < Z < 0.4573)

From the Z-table, the probability for Z = -1.4497 is approximately 0.0735, and the probability for Z = 0.4573 is approximately 0.6745.

Therefore, P(934.8 < X < 965.2) ≈ 0.6745 - 0.0735 ≈ 0.6010.

2. Probability for a random sample of size n=99 (x̄):

For the sample mean, the standard error (SE) is given by SE = σ / √n = 144 / √99 ≈ 14.4694.

We find the z-scores for the values 934.8 and 965.2 using the sample mean:

Z1 = (934.8 - 958) / 14.4694 ≈ -1.6374 Z2 = (965.2 - 958) / 14.4694 ≈ 0.4849

Now, we find the probabilities using the Z-table: P(934.8 < x̄ < 965.2) = P(-1.6374 < Z < 0.4849)

From the Z-table, the probability for Z = -1.6374 is approximately 0.0500, and the probability for Z = 0.4849 is approximately 0.6868.

Therefore, P(934.8 < x̄ < 965.2) ≈ 0.6868 - 0.0500 ≈ 0.6368.

Answers:

1. P(934.8 < X < 965.2) ≈ 0.6010

2. P(934.8 < x̄ < 965.2) ≈ 0.6368