Answer :
To solve the problem of finding [tex]\( P(120 \leq X \leq 153) \)[/tex] where the weight [tex]\( X \)[/tex] is a normally distributed random variable with a mean of 147 pounds and a standard deviation of 16 pounds, we can follow these steps:
1. Understand the Problem: We want to find the probability that the weight [tex]\( X \)[/tex] is between 120 and 153 pounds. This requires calculating the area under the normal distribution curve between these two values.
2. Standardize the Variable: Since [tex]\( X \)[/tex] is normally distributed, we use the z-score formula to standardize the values. The z-score formula is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
3. Calculate the Z-scores:
- For the lower bound [tex]\( X = 120 \)[/tex]:
[tex]\[
z_{\text{lower}} = \frac{120 - 147}{16} = -1.6875
\][/tex]
- For the upper bound [tex]\( X = 153 \)[/tex]:
[tex]\[
z_{\text{upper}} = \frac{153 - 147}{16} = 0.375
\][/tex]
4. Use the Standard Normal Distribution: With the z-scores calculated, we now use the standard normal distribution (z-table or a computational tool) to find the cumulative probabilities:
- Cumulative probability for [tex]\( z_{\text{lower}} = -1.6875 \)[/tex].
- Cumulative probability for [tex]\( z_{\text{upper}} = 0.375 \)[/tex].
5. Find the Probability: The probability that [tex]\( X \)[/tex] is between 120 and 153 is the difference between these cumulative probabilities:
[tex]\[
P(120 \leq X \leq 153) = P(z_{\text{upper}}) - P(z_{\text{lower}})
\][/tex]
6. Result: After finding the cumulative probabilities from the standard normal distribution, the probability that [tex]\( X \)[/tex] is between 120 and 153 pounds comes out to be approximately 0.6004.
Therefore, the probability that a 40-year-old man weighs between 120 and 153 pounds is about 0.6004, or 60.04%.
1. Understand the Problem: We want to find the probability that the weight [tex]\( X \)[/tex] is between 120 and 153 pounds. This requires calculating the area under the normal distribution curve between these two values.
2. Standardize the Variable: Since [tex]\( X \)[/tex] is normally distributed, we use the z-score formula to standardize the values. The z-score formula is:
[tex]\[
z = \frac{X - \mu}{\sigma}
\][/tex]
where [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
3. Calculate the Z-scores:
- For the lower bound [tex]\( X = 120 \)[/tex]:
[tex]\[
z_{\text{lower}} = \frac{120 - 147}{16} = -1.6875
\][/tex]
- For the upper bound [tex]\( X = 153 \)[/tex]:
[tex]\[
z_{\text{upper}} = \frac{153 - 147}{16} = 0.375
\][/tex]
4. Use the Standard Normal Distribution: With the z-scores calculated, we now use the standard normal distribution (z-table or a computational tool) to find the cumulative probabilities:
- Cumulative probability for [tex]\( z_{\text{lower}} = -1.6875 \)[/tex].
- Cumulative probability for [tex]\( z_{\text{upper}} = 0.375 \)[/tex].
5. Find the Probability: The probability that [tex]\( X \)[/tex] is between 120 and 153 is the difference between these cumulative probabilities:
[tex]\[
P(120 \leq X \leq 153) = P(z_{\text{upper}}) - P(z_{\text{lower}})
\][/tex]
6. Result: After finding the cumulative probabilities from the standard normal distribution, the probability that [tex]\( X \)[/tex] is between 120 and 153 pounds comes out to be approximately 0.6004.
Therefore, the probability that a 40-year-old man weighs between 120 and 153 pounds is about 0.6004, or 60.04%.
