Answer :
Sure! Let's walk through the solution step by step.
### Problem a:
You need to find the probability that a single randomly selected value from a normal distribution is between 51.7 and 52.4. The distribution has a mean [tex]\(\mu = 51\)[/tex] and a standard deviation [tex]\(\sigma = 50\)[/tex].
1. Calculate the z-scores for the bounds:
- For the lower bound (51.7):
[tex]\[
z_{\text{lower}} = \frac{51.7 - 51}{50}
\][/tex]
This calculates the z-score for 51.7.
- For the upper bound (52.4):
[tex]\[
z_{\text{upper}} = \frac{52.4 - 51}{50}
\][/tex]
This calculates the z-score for 52.4.
2. Find the probabilities using the standard normal distribution:
- Use the cumulative distribution function (CDF) to find the probability corresponding to each z-score.
- The probability that a value is between 51.7 and 52.4 is the difference between the CDF values at these z-scores:
[tex]\[
P(51.7 < X < 52.4) = \text{CDF}(z_{\text{upper}}) - \text{CDF}(z_{\text{lower}})
\][/tex]
After performing these calculations, the probability that a single value is between 51.7 and 52.4 is approximately 0.0056.
### Problem b:
Now, you need to find the probability that the mean of a sample of size [tex]\(n = 49\)[/tex] is between 51.7 and 52.4.
1. Calculate the standard deviation of the sample mean (standard error):
[tex]\[
\sigma_{\text{sample}} = \frac{50}{\sqrt{49}}
\][/tex]
This is because the standard deviation of the sample mean ([tex]\(\sigma_{\text{sample}}\)[/tex]) is the population standard deviation divided by the square root of the sample size.
2. Calculate the z-scores for the bounds based on the sample:
- For the lower bound (51.7):
[tex]\[
z_{\text{lower sample}} = \frac{51.7 - 51}{\sigma_{\text{sample}}}
\][/tex]
This calculates the z-score for the sample mean lower bound.
- For the upper bound (52.4):
[tex]\[
z_{\text{upper sample}} = \frac{52.4 - 51}{\sigma_{\text{sample}}}
\][/tex]
This calculates the z-score for the sample mean upper bound.
3. Find the probabilities for the sample mean using the standard normal distribution:
- Again, use the CDF to determine the probabilities for these z-scores.
- The probability that the sample mean is between 51.7 and 52.4 is:
[tex]\[
P(51.7 < \bar{X} < 52.4) = \text{CDF}(z_{\text{upper sample}}) - \text{CDF}(z_{\text{lower sample}})
\][/tex]
After performing these calculations, the probability that the sample mean is between 51.7 and 52.4 is approximately 0.0387.
These calculations show how likely it is for a value, or a sample mean, to fall within a specific range given a normal distribution.
### Problem a:
You need to find the probability that a single randomly selected value from a normal distribution is between 51.7 and 52.4. The distribution has a mean [tex]\(\mu = 51\)[/tex] and a standard deviation [tex]\(\sigma = 50\)[/tex].
1. Calculate the z-scores for the bounds:
- For the lower bound (51.7):
[tex]\[
z_{\text{lower}} = \frac{51.7 - 51}{50}
\][/tex]
This calculates the z-score for 51.7.
- For the upper bound (52.4):
[tex]\[
z_{\text{upper}} = \frac{52.4 - 51}{50}
\][/tex]
This calculates the z-score for 52.4.
2. Find the probabilities using the standard normal distribution:
- Use the cumulative distribution function (CDF) to find the probability corresponding to each z-score.
- The probability that a value is between 51.7 and 52.4 is the difference between the CDF values at these z-scores:
[tex]\[
P(51.7 < X < 52.4) = \text{CDF}(z_{\text{upper}}) - \text{CDF}(z_{\text{lower}})
\][/tex]
After performing these calculations, the probability that a single value is between 51.7 and 52.4 is approximately 0.0056.
### Problem b:
Now, you need to find the probability that the mean of a sample of size [tex]\(n = 49\)[/tex] is between 51.7 and 52.4.
1. Calculate the standard deviation of the sample mean (standard error):
[tex]\[
\sigma_{\text{sample}} = \frac{50}{\sqrt{49}}
\][/tex]
This is because the standard deviation of the sample mean ([tex]\(\sigma_{\text{sample}}\)[/tex]) is the population standard deviation divided by the square root of the sample size.
2. Calculate the z-scores for the bounds based on the sample:
- For the lower bound (51.7):
[tex]\[
z_{\text{lower sample}} = \frac{51.7 - 51}{\sigma_{\text{sample}}}
\][/tex]
This calculates the z-score for the sample mean lower bound.
- For the upper bound (52.4):
[tex]\[
z_{\text{upper sample}} = \frac{52.4 - 51}{\sigma_{\text{sample}}}
\][/tex]
This calculates the z-score for the sample mean upper bound.
3. Find the probabilities for the sample mean using the standard normal distribution:
- Again, use the CDF to determine the probabilities for these z-scores.
- The probability that the sample mean is between 51.7 and 52.4 is:
[tex]\[
P(51.7 < \bar{X} < 52.4) = \text{CDF}(z_{\text{upper sample}}) - \text{CDF}(z_{\text{lower sample}})
\][/tex]
After performing these calculations, the probability that the sample mean is between 51.7 and 52.4 is approximately 0.0387.
These calculations show how likely it is for a value, or a sample mean, to fall within a specific range given a normal distribution.
