Answer :
To solve this problem, we need to use the properties of the sampling distribution of the sample mean. When a random sample is taken from a population with a known mean and standard deviation, the sample mean follows a normal distribution if the sample size is sufficiently large. This distribution has a mean equal to the population mean and a standard deviation called the standard error, calculated as the population standard deviation divided by the square root of the sample size.
Let's go through each part step-by-step:
### Given:
- Population Mean ([tex]\(\mu\)[/tex]) = 50
- Population Standard Deviation ([tex]\(\sigma\)[/tex]) = 18
- Sample Size ([tex]\(n\)[/tex]) = 64
### Standard Error:
The standard error (SE) is calculated using the formula:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{18}{\sqrt{64}} = \frac{18}{8} = 2.25 \][/tex]
Now, let's calculate each probability:
### a. Probability that the sample mean is greater than 54:
1. Calculate the Z-score:
[tex]\[ z = \frac{54 - 50}{2.25} = \frac{4}{2.25} \approx 1.78 \][/tex]
2. Use the standard normal distribution to find the probability of Z being greater than 1.78. This is equivalent to:
[tex]\[ P(\bar{x} > 54) = 1 - P(Z < 1.78) = 0.0377 \][/tex]
### b. Probability that the sample mean is less than 54:
The probability that the sample mean is less than 54 is:
[tex]\[ P(\bar{x} < 54) = P(Z < 1.78) = 0.9623 \][/tex]
### c. Probability that the sample mean is less than 49:
1. Calculate the Z-score:
[tex]\[ z = \frac{49 - 50}{2.25} = \frac{-1}{2.25} \approx -0.44 \][/tex]
2. Find the probability:
[tex]\[ P(\bar{x} < 49) = P(Z < -0.44) = 0.3284 \][/tex]
### d. Probability that the sample mean is between 48.5 and 53.5:
1. Calculate the Z-scores for 48.5 and 53.5:
[tex]\[ z_1 = \frac{48.5 - 50}{2.25} \approx -0.67 \][/tex]
[tex]\[ z_2 = \frac{53.5 - 50}{2.25} \approx 1.56 \][/tex]
2. Calculate the probability:
[tex]\[ P(48.5 \leq \bar{x} \leq 53.5) = P(-0.67 \leq Z \leq 1.56) = P(Z < 1.56) - P(Z < -0.67) = 0.6876 \][/tex]
### e. Probability that the sample mean is between 50.1 and 51.7:
1. Calculate the Z-scores for 50.1 and 51.7:
[tex]\[ z_1 = \frac{50.1 - 50}{2.25} \approx 0.04 \][/tex]
[tex]\[ z_2 = \frac{51.7 - 50}{2.25} \approx 0.76 \][/tex]
2. Calculate the probability:
[tex]\[ P(50.1 \leq \bar{x} \leq 51.7) = P(0.04 \leq Z \leq 0.76) = P(Z < 0.76) - P(Z < 0.04) = 0.2573 \][/tex]
These calculations give us the probabilities for each scenario requested in the problem.
Let's go through each part step-by-step:
### Given:
- Population Mean ([tex]\(\mu\)[/tex]) = 50
- Population Standard Deviation ([tex]\(\sigma\)[/tex]) = 18
- Sample Size ([tex]\(n\)[/tex]) = 64
### Standard Error:
The standard error (SE) is calculated using the formula:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{18}{\sqrt{64}} = \frac{18}{8} = 2.25 \][/tex]
Now, let's calculate each probability:
### a. Probability that the sample mean is greater than 54:
1. Calculate the Z-score:
[tex]\[ z = \frac{54 - 50}{2.25} = \frac{4}{2.25} \approx 1.78 \][/tex]
2. Use the standard normal distribution to find the probability of Z being greater than 1.78. This is equivalent to:
[tex]\[ P(\bar{x} > 54) = 1 - P(Z < 1.78) = 0.0377 \][/tex]
### b. Probability that the sample mean is less than 54:
The probability that the sample mean is less than 54 is:
[tex]\[ P(\bar{x} < 54) = P(Z < 1.78) = 0.9623 \][/tex]
### c. Probability that the sample mean is less than 49:
1. Calculate the Z-score:
[tex]\[ z = \frac{49 - 50}{2.25} = \frac{-1}{2.25} \approx -0.44 \][/tex]
2. Find the probability:
[tex]\[ P(\bar{x} < 49) = P(Z < -0.44) = 0.3284 \][/tex]
### d. Probability that the sample mean is between 48.5 and 53.5:
1. Calculate the Z-scores for 48.5 and 53.5:
[tex]\[ z_1 = \frac{48.5 - 50}{2.25} \approx -0.67 \][/tex]
[tex]\[ z_2 = \frac{53.5 - 50}{2.25} \approx 1.56 \][/tex]
2. Calculate the probability:
[tex]\[ P(48.5 \leq \bar{x} \leq 53.5) = P(-0.67 \leq Z \leq 1.56) = P(Z < 1.56) - P(Z < -0.67) = 0.6876 \][/tex]
### e. Probability that the sample mean is between 50.1 and 51.7:
1. Calculate the Z-scores for 50.1 and 51.7:
[tex]\[ z_1 = \frac{50.1 - 50}{2.25} \approx 0.04 \][/tex]
[tex]\[ z_2 = \frac{51.7 - 50}{2.25} \approx 0.76 \][/tex]
2. Calculate the probability:
[tex]\[ P(50.1 \leq \bar{x} \leq 51.7) = P(0.04 \leq Z \leq 0.76) = P(Z < 0.76) - P(Z < 0.04) = 0.2573 \][/tex]
These calculations give us the probabilities for each scenario requested in the problem.
