Answer :
To solve the problem, we need to find the probabilities related to a normally distributed population with a mean [tex]\(\mu = 40\)[/tex] and a variance [tex]\(\sigma^2 = 64\)[/tex]. The standard deviation [tex]\(\sigma\)[/tex] is the square root of the variance, which is [tex]\(\sigma = \sqrt{64} = 8\)[/tex].
### i) Probability that it is greater than 55
First, we find the z-score for 55. The z-score tells us how many standard deviations away 55 is from the mean.
[tex]\[ z = \frac{X - \mu}{\sigma} = \frac{55 - 40}{8} = 1.875 \][/tex]
We then look up the z-score in a standard normal distribution table or use a standard normal distribution calculator to find the probability that a value is less than 1.875. This gives us the cumulative distribution function (CDF) value.
[tex]\[ P(Z < 1.875) = \text{CDF}(1.875) \approx 0.9696 \][/tex]
To find the probability that a value is greater than 55, we take the complement of this CDF value:
[tex]\[ P(X > 55) = 1 - \text{CDF}(1.875) = 1 - 0.9696 = 0.0304 \approx 0.030396361765261393 \][/tex]
So, the probability that it is greater than 55 is approximately [tex]\(0.0304\)[/tex].
### ii) Probability that it is less than 32
Next, we find the z-score for 32:
[tex]\[ z = \frac{32 - 40}{8} = -1.0 \][/tex]
We then look up the z-score in a standard normal distribution table or use a standard normal distribution calculator to find the CDF value for [tex]\(-1.0\)[/tex]:
[tex]\[ P(Z < -1.0) = \text{CDF}(-1.0) \approx 0.1587 \approx 0.15865525393145707 \][/tex]
Therefore, the probability that it is less than 32 is approximately [tex]\(0.1587\)[/tex].
### iii) Probability that it is between 35 and 50
We need to find the probability that it is between 35 and 50. This involves finding two z-scores and calculating the difference between their CDF values.
First, find the z-score for 35:
[tex]\[ z_{35} = \frac{35 - 40}{8} = -0.625 \][/tex]
Next, find the z-score for 50:
[tex]\[ z_{50} = \frac{50 - 40}{8} = 1.25 \][/tex]
Now, we find the CDF values for these z-scores:
[tex]\[ P(Z < -0.625) = \text{CDF}(-0.625) \approx 0.266 \][/tex]
[tex]\[ P(Z < 1.25) = \text{CDF}(1.25) \approx 0.8944 \][/tex]
To find the probability that a value is between 35 and 50, we subtract the CDF value for [tex]\(z_{35}\)[/tex] from the CDF value for [tex]\(z_{50}\)[/tex]:
[tex]\[ P(35 < X < 50) = \text{CDF}(1.25) - \text{CDF}(-0.625) = 0.8944 - 0.266 = 0.6284 \approx 0.6283646972844441 \][/tex]
Thus, the probability that it is between 35 and 50 is approximately [tex]\(0.6284\)[/tex].
### Summary
- Probability that it is greater than 55: [tex]\( \approx 0.0304 \)[/tex]
- Probability that it is less than 32: [tex]\( \approx 0.1587 \)[/tex]
- Probability that it is between 35 and 50: [tex]\( \approx 0.6284 \)[/tex]
### i) Probability that it is greater than 55
First, we find the z-score for 55. The z-score tells us how many standard deviations away 55 is from the mean.
[tex]\[ z = \frac{X - \mu}{\sigma} = \frac{55 - 40}{8} = 1.875 \][/tex]
We then look up the z-score in a standard normal distribution table or use a standard normal distribution calculator to find the probability that a value is less than 1.875. This gives us the cumulative distribution function (CDF) value.
[tex]\[ P(Z < 1.875) = \text{CDF}(1.875) \approx 0.9696 \][/tex]
To find the probability that a value is greater than 55, we take the complement of this CDF value:
[tex]\[ P(X > 55) = 1 - \text{CDF}(1.875) = 1 - 0.9696 = 0.0304 \approx 0.030396361765261393 \][/tex]
So, the probability that it is greater than 55 is approximately [tex]\(0.0304\)[/tex].
### ii) Probability that it is less than 32
Next, we find the z-score for 32:
[tex]\[ z = \frac{32 - 40}{8} = -1.0 \][/tex]
We then look up the z-score in a standard normal distribution table or use a standard normal distribution calculator to find the CDF value for [tex]\(-1.0\)[/tex]:
[tex]\[ P(Z < -1.0) = \text{CDF}(-1.0) \approx 0.1587 \approx 0.15865525393145707 \][/tex]
Therefore, the probability that it is less than 32 is approximately [tex]\(0.1587\)[/tex].
### iii) Probability that it is between 35 and 50
We need to find the probability that it is between 35 and 50. This involves finding two z-scores and calculating the difference between their CDF values.
First, find the z-score for 35:
[tex]\[ z_{35} = \frac{35 - 40}{8} = -0.625 \][/tex]
Next, find the z-score for 50:
[tex]\[ z_{50} = \frac{50 - 40}{8} = 1.25 \][/tex]
Now, we find the CDF values for these z-scores:
[tex]\[ P(Z < -0.625) = \text{CDF}(-0.625) \approx 0.266 \][/tex]
[tex]\[ P(Z < 1.25) = \text{CDF}(1.25) \approx 0.8944 \][/tex]
To find the probability that a value is between 35 and 50, we subtract the CDF value for [tex]\(z_{35}\)[/tex] from the CDF value for [tex]\(z_{50}\)[/tex]:
[tex]\[ P(35 < X < 50) = \text{CDF}(1.25) - \text{CDF}(-0.625) = 0.8944 - 0.266 = 0.6284 \approx 0.6283646972844441 \][/tex]
Thus, the probability that it is between 35 and 50 is approximately [tex]\(0.6284\)[/tex].
### Summary
- Probability that it is greater than 55: [tex]\( \approx 0.0304 \)[/tex]
- Probability that it is less than 32: [tex]\( \approx 0.1587 \)[/tex]
- Probability that it is between 35 and 50: [tex]\( \approx 0.6284 \)[/tex]
