Answer :
Certainly! Let’s break down the calculation of the probabilities step-by-step for a normally distributed population with a mean ([tex]\(\mu\)[/tex]) of 40 and variance ([tex]\(\sigma^2\)[/tex]) of 64. The standard deviation ([tex]\(\sigma\)[/tex]) would then be the square root of the variance, which is 8.
### i) Probability that [tex]\(x\)[/tex] is greater than 55
1. Finding the z-score:
The z-score tells us how many standard deviations away the value is from the mean. The formula for the z-score is:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
Plugging in [tex]\(x = 55\)[/tex]:
[tex]\[ z = \frac{55 - 40}{8} = \frac{15}{8} = 1.875 \][/tex]
2. Finding the probability:
To find the probability that [tex]\(x\)[/tex] is greater than 55, we need to look up the z-score in the standard normal distribution table, or use a cumulative distribution function (CDF).
The cumulative probability up to [tex]\(z = 1.875\)[/tex] can be found using a standard normal distribution table or calculator. This gives us a probability [tex]\(P(Z \leq 1.875)\)[/tex].
3. Calculating the probability:
Since we want the probability that [tex]\(x\)[/tex] is greater than 55, we need [tex]\(1 - P(Z \leq 1.875)\)[/tex]:
[tex]\[ P(X > 55) = 1 - P(Z \leq 1.875) \approx 1 - 0.9696 = 0.0304 \][/tex]
So, the probability that [tex]\(x\)[/tex] is greater than 55 is approximately 0.0304.
### ii) Probability that [tex]\(x\)[/tex] is less than 32
1. Finding the z-score:
Using the same z-score formula:
[tex]\[ z = \frac{32 - 40}{8} = \frac{-8}{8} = -1 \][/tex]
2. Finding the probability:
We look up the cumulative probability up to [tex]\(z = -1\)[/tex] in the standard normal distribution table or use a CDF function.
3. Calculating the probability:
The cumulative probability [tex]\(P(Z \leq -1)\)[/tex]:
[tex]\[ P(X < 32) = P(Z \leq -1) \approx 0.1587 \][/tex]
So, the probability that [tex]\(x\)[/tex] is less than 32 is approximately 0.1587.
### iii) Probability that [tex]\(x\)[/tex] is between 35 and 50
1. Finding the z-scores:
For [tex]\(x = 35\)[/tex]:
[tex]\[ z = \frac{35 - 40}{8} = \frac{-5}{8} = -0.625 \][/tex]
For [tex]\(x = 50\)[/tex]:
[tex]\[ z = \frac{50 - 40}{8} = \frac{10}{8} = 1.25 \][/tex]
2. Finding the probabilities:
We look up the cumulative probabilities:
[tex]\[ P(Z \leq -0.625) \quad \text{and} \quad P(Z \leq 1.25) \][/tex]
3. Calculating the probability:
The desired probability is the difference between these two cumulative probabilities:
[tex]\[ P(35 \leq X \leq 50) = P(Z \leq 1.25) - P(Z \leq -0.625) \approx 0.8944 - 0.2660 = 0.6284 \][/tex]
So, the probability that [tex]\(x\)[/tex] is between 35 and 50 is approximately 0.6284.
### Summary of Results:
- The probability that [tex]\(x\)[/tex] is greater than 55 is approximately 0.0304.
- The probability that [tex]\(x\)[/tex] is less than 32 is approximately 0.1587.
- The probability that [tex]\(x\)[/tex] is between 35 and 50 is approximately 0.6284.
These results are consistent with the rules and characteristics of the normal distribution and provide the needed probabilities for the given scenarios.
### i) Probability that [tex]\(x\)[/tex] is greater than 55
1. Finding the z-score:
The z-score tells us how many standard deviations away the value is from the mean. The formula for the z-score is:
[tex]\[ z = \frac{x - \mu}{\sigma} \][/tex]
Plugging in [tex]\(x = 55\)[/tex]:
[tex]\[ z = \frac{55 - 40}{8} = \frac{15}{8} = 1.875 \][/tex]
2. Finding the probability:
To find the probability that [tex]\(x\)[/tex] is greater than 55, we need to look up the z-score in the standard normal distribution table, or use a cumulative distribution function (CDF).
The cumulative probability up to [tex]\(z = 1.875\)[/tex] can be found using a standard normal distribution table or calculator. This gives us a probability [tex]\(P(Z \leq 1.875)\)[/tex].
3. Calculating the probability:
Since we want the probability that [tex]\(x\)[/tex] is greater than 55, we need [tex]\(1 - P(Z \leq 1.875)\)[/tex]:
[tex]\[ P(X > 55) = 1 - P(Z \leq 1.875) \approx 1 - 0.9696 = 0.0304 \][/tex]
So, the probability that [tex]\(x\)[/tex] is greater than 55 is approximately 0.0304.
### ii) Probability that [tex]\(x\)[/tex] is less than 32
1. Finding the z-score:
Using the same z-score formula:
[tex]\[ z = \frac{32 - 40}{8} = \frac{-8}{8} = -1 \][/tex]
2. Finding the probability:
We look up the cumulative probability up to [tex]\(z = -1\)[/tex] in the standard normal distribution table or use a CDF function.
3. Calculating the probability:
The cumulative probability [tex]\(P(Z \leq -1)\)[/tex]:
[tex]\[ P(X < 32) = P(Z \leq -1) \approx 0.1587 \][/tex]
So, the probability that [tex]\(x\)[/tex] is less than 32 is approximately 0.1587.
### iii) Probability that [tex]\(x\)[/tex] is between 35 and 50
1. Finding the z-scores:
For [tex]\(x = 35\)[/tex]:
[tex]\[ z = \frac{35 - 40}{8} = \frac{-5}{8} = -0.625 \][/tex]
For [tex]\(x = 50\)[/tex]:
[tex]\[ z = \frac{50 - 40}{8} = \frac{10}{8} = 1.25 \][/tex]
2. Finding the probabilities:
We look up the cumulative probabilities:
[tex]\[ P(Z \leq -0.625) \quad \text{and} \quad P(Z \leq 1.25) \][/tex]
3. Calculating the probability:
The desired probability is the difference between these two cumulative probabilities:
[tex]\[ P(35 \leq X \leq 50) = P(Z \leq 1.25) - P(Z \leq -0.625) \approx 0.8944 - 0.2660 = 0.6284 \][/tex]
So, the probability that [tex]\(x\)[/tex] is between 35 and 50 is approximately 0.6284.
### Summary of Results:
- The probability that [tex]\(x\)[/tex] is greater than 55 is approximately 0.0304.
- The probability that [tex]\(x\)[/tex] is less than 32 is approximately 0.1587.
- The probability that [tex]\(x\)[/tex] is between 35 and 50 is approximately 0.6284.
These results are consistent with the rules and characteristics of the normal distribution and provide the needed probabilities for the given scenarios.
