Answer :
Certainly, let's carefully solve the given problem step-by-step:
We are given the following information:
- Population mean pulse rate, [tex]\(\mu = 72.0\)[/tex] beats per minute.
- Population standard deviation, [tex]\(\sigma = 12.5\)[/tex] beats per minute.
### Part (a):
If 1 adult female is randomly selected, we need to find the probability that her pulse rate is between 65 beats per minute and 79 beats per minute.
1. Standardize the bounds:
We will use the Z-score formula to convert the bounds to standard normal distribution values.
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
For the lower bound (65 beats per minute):
[tex]\[ Z_{\text{lower}} = \frac{65 - 72.0}{12.5} = \frac{-7}{12.5} = -0.56 \][/tex]
For the upper bound (79 beats per minute):
[tex]\[ Z_{\text{upper}} = \frac{79 - 72.0}{12.5} = \frac{7}{12.5} = 0.56 \][/tex]
2. Find the probability corresponding to each Z-score:
Using the standard normal distribution table (or a calculator for the cumulative distribution function):
[tex]\[ P(Z < 0.56) - P(Z < -0.56) \][/tex]
3. Probability Calculation:
- [tex]\( P(Z < 0.56) \approx 0.7123 \)[/tex]
- [tex]\( P(Z < -0.56) \approx 0.2878 \)[/tex]
- Therefore, the probability is:
[tex]\[ P(65 < X < 79) = 0.7123 - 0.2878 = 0.4245 \][/tex]
So, the probability that one randomly selected adult female has a pulse rate between 65 and 79 beats per minute is [tex]\(0.4245\)[/tex].
### Part (b):
If a sample of 4 adult females is randomly selected, we need to find the probability that their sample mean pulse rate is between 65 beats per minute and 79 beats per minute.
1. Standard Error of the Mean:
The standard error of the mean (SEM) for a sample size of [tex]\( n \)[/tex] is calculated as:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{12.5}{\sqrt{4}} = \frac{12.5}{2} = 6.25 \][/tex]
2. Standardize the bounds:
For the lower bound (65 beats per minute):
[tex]\[ Z_{\text{lower}} = \frac{65 - 72.0}{6.25} = \frac{-7}{6.25} = -1.12 \][/tex]
For the upper bound (79 beats per minute):
[tex]\[ Z_{\text{upper}} = \frac{79 - 72.0}{6.25} = \frac{7}{6.25} = 1.12 \][/tex]
3. Find the probability corresponding to each Z-score:
Using the standard normal distribution table (or a calculator for the cumulative distribution function):
[tex]\[ P(Z < 1.12) - P(Z < -1.12) \][/tex]
4. Probability Calculation:
- [tex]\( P(Z < 1.12) \approx 0.8686 \)[/tex]
- [tex]\( P(Z < -1.12) \approx 0.1314 \)[/tex]
- Therefore, the probability is:
[tex]\[ P(65 < \overline{X} < 79) = 0.8686 - 0.1314 = 0.7373 \][/tex]
So, the probability that the sample mean pulse rate of 4 randomly selected adult females is between 65 and 79 beats per minute is [tex]\(0.7373\)[/tex].
In summary:
- The probability for one adult female is [tex]\(0.4245\)[/tex].
- The probability for the sample mean of 4 adult females is [tex]\(0.7373\)[/tex].
We are given the following information:
- Population mean pulse rate, [tex]\(\mu = 72.0\)[/tex] beats per minute.
- Population standard deviation, [tex]\(\sigma = 12.5\)[/tex] beats per minute.
### Part (a):
If 1 adult female is randomly selected, we need to find the probability that her pulse rate is between 65 beats per minute and 79 beats per minute.
1. Standardize the bounds:
We will use the Z-score formula to convert the bounds to standard normal distribution values.
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
For the lower bound (65 beats per minute):
[tex]\[ Z_{\text{lower}} = \frac{65 - 72.0}{12.5} = \frac{-7}{12.5} = -0.56 \][/tex]
For the upper bound (79 beats per minute):
[tex]\[ Z_{\text{upper}} = \frac{79 - 72.0}{12.5} = \frac{7}{12.5} = 0.56 \][/tex]
2. Find the probability corresponding to each Z-score:
Using the standard normal distribution table (or a calculator for the cumulative distribution function):
[tex]\[ P(Z < 0.56) - P(Z < -0.56) \][/tex]
3. Probability Calculation:
- [tex]\( P(Z < 0.56) \approx 0.7123 \)[/tex]
- [tex]\( P(Z < -0.56) \approx 0.2878 \)[/tex]
- Therefore, the probability is:
[tex]\[ P(65 < X < 79) = 0.7123 - 0.2878 = 0.4245 \][/tex]
So, the probability that one randomly selected adult female has a pulse rate between 65 and 79 beats per minute is [tex]\(0.4245\)[/tex].
### Part (b):
If a sample of 4 adult females is randomly selected, we need to find the probability that their sample mean pulse rate is between 65 beats per minute and 79 beats per minute.
1. Standard Error of the Mean:
The standard error of the mean (SEM) for a sample size of [tex]\( n \)[/tex] is calculated as:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{12.5}{\sqrt{4}} = \frac{12.5}{2} = 6.25 \][/tex]
2. Standardize the bounds:
For the lower bound (65 beats per minute):
[tex]\[ Z_{\text{lower}} = \frac{65 - 72.0}{6.25} = \frac{-7}{6.25} = -1.12 \][/tex]
For the upper bound (79 beats per minute):
[tex]\[ Z_{\text{upper}} = \frac{79 - 72.0}{6.25} = \frac{7}{6.25} = 1.12 \][/tex]
3. Find the probability corresponding to each Z-score:
Using the standard normal distribution table (or a calculator for the cumulative distribution function):
[tex]\[ P(Z < 1.12) - P(Z < -1.12) \][/tex]
4. Probability Calculation:
- [tex]\( P(Z < 1.12) \approx 0.8686 \)[/tex]
- [tex]\( P(Z < -1.12) \approx 0.1314 \)[/tex]
- Therefore, the probability is:
[tex]\[ P(65 < \overline{X} < 79) = 0.8686 - 0.1314 = 0.7373 \][/tex]
So, the probability that the sample mean pulse rate of 4 randomly selected adult females is between 65 and 79 beats per minute is [tex]\(0.7373\)[/tex].
In summary:
- The probability for one adult female is [tex]\(0.4245\)[/tex].
- The probability for the sample mean of 4 adult females is [tex]\(0.7373\)[/tex].
