Answer :
To construct a 98% confidence interval estimate of the standard deviation of body temperature of all healthy humans, we will follow these steps:
1. Identify the known sample statistics and level of confidence:
- Sample size [tex]\( n = 109 \)[/tex]
- Sample mean [tex]\( \bar{x} = 98.30^\circ F \)[/tex] (though not directly needed for the confidence interval for the standard deviation, it generally helps provide context)
- Sample standard deviation [tex]\( s = 0.63^\circ F \)[/tex]
- Confidence level [tex]\( 0.98 \)[/tex]
2. Calculate the degrees of freedom:
The degrees of freedom ([tex]\( df \)[/tex]) is calculated as:
[tex]\[ df = n - 1 = 109 - 1 = 108 \][/tex]
3. Determine the alpha level and its respective tail areas:
Alpha (α) is the significance level, which is [tex]\( 1 - \text{confidence level} \)[/tex]:
[tex]\[ \alpha = 1 - 0.98 = 0.02 \][/tex]
Since we are dealing with a two-tailed test, we split the alpha into two equal parts:
[tex]\[ \alpha / 2 = 0.02 / 2 = 0.01 \][/tex]
4. Find the critical values from the Chi-Square distribution:
Using the Chi-Square distribution table and the degrees of freedom ([tex]\( df = 108 \)[/tex]), find the critical values corresponding to [tex]\( \alpha/2 \)[/tex] and [tex]\( 1 - \alpha/2 \)[/tex].
- [tex]\( \chi^2_{(\alpha/2)} = \chi^2_{(0.01, 108)} \)[/tex]
- [tex]\( \chi^2_{(1 - \alpha/2)} = \chi^2_{(0.99, 108)} \)[/tex]
The critical values from the Chi-Square table are:
- [tex]\( \chi^2_{(0.01, 108)} \)[/tex] is approximately 74.222
- [tex]\( \chi^2_{(0.99, 108)} \)[/tex] is approximately 146.567
5. Calculate the lower and upper bounds of the confidence interval:
The formula for the confidence interval for the population standard deviation [tex]\( \sigma \)[/tex] is:
[tex]\[ \left( \sqrt{ \frac{(n-1)s^2}{\chi^2_{(\text{upper})}} }, \sqrt{ \frac{(n-1)s^2}{\chi^2_{(\text{lower})}} } \right) \][/tex]
Plugging in the values:
[tex]\[ \text{Lower bound} = \sqrt{ \frac{(109-1) \cdot (0.63)^2}{146.567} } = \sqrt{ \frac{108 \cdot 0.3969}{146.567} } = \sqrt{ \frac{42.0642}{146.567} } \approx 0.54 \][/tex]
[tex]\[ \text{Upper bound} = \sqrt{ \frac{(109-1) \cdot (0.63)^2}{74.222} } = \sqrt{ \frac{108 \cdot 0.3969}{74.222} } = \sqrt{ \frac{42.0642}{74.222} } \approx 0.75 \][/tex]
6. State the final confidence interval:
Therefore, the 98% confidence interval estimate for the standard deviation of body temperature of all healthy humans is:
[tex]\[ 0.54^\circ \text{F} < \sigma < 0.75^\circ \text{F} \][/tex]
Make sure to round your final answers to two decimal places as needed.
1. Identify the known sample statistics and level of confidence:
- Sample size [tex]\( n = 109 \)[/tex]
- Sample mean [tex]\( \bar{x} = 98.30^\circ F \)[/tex] (though not directly needed for the confidence interval for the standard deviation, it generally helps provide context)
- Sample standard deviation [tex]\( s = 0.63^\circ F \)[/tex]
- Confidence level [tex]\( 0.98 \)[/tex]
2. Calculate the degrees of freedom:
The degrees of freedom ([tex]\( df \)[/tex]) is calculated as:
[tex]\[ df = n - 1 = 109 - 1 = 108 \][/tex]
3. Determine the alpha level and its respective tail areas:
Alpha (α) is the significance level, which is [tex]\( 1 - \text{confidence level} \)[/tex]:
[tex]\[ \alpha = 1 - 0.98 = 0.02 \][/tex]
Since we are dealing with a two-tailed test, we split the alpha into two equal parts:
[tex]\[ \alpha / 2 = 0.02 / 2 = 0.01 \][/tex]
4. Find the critical values from the Chi-Square distribution:
Using the Chi-Square distribution table and the degrees of freedom ([tex]\( df = 108 \)[/tex]), find the critical values corresponding to [tex]\( \alpha/2 \)[/tex] and [tex]\( 1 - \alpha/2 \)[/tex].
- [tex]\( \chi^2_{(\alpha/2)} = \chi^2_{(0.01, 108)} \)[/tex]
- [tex]\( \chi^2_{(1 - \alpha/2)} = \chi^2_{(0.99, 108)} \)[/tex]
The critical values from the Chi-Square table are:
- [tex]\( \chi^2_{(0.01, 108)} \)[/tex] is approximately 74.222
- [tex]\( \chi^2_{(0.99, 108)} \)[/tex] is approximately 146.567
5. Calculate the lower and upper bounds of the confidence interval:
The formula for the confidence interval for the population standard deviation [tex]\( \sigma \)[/tex] is:
[tex]\[ \left( \sqrt{ \frac{(n-1)s^2}{\chi^2_{(\text{upper})}} }, \sqrt{ \frac{(n-1)s^2}{\chi^2_{(\text{lower})}} } \right) \][/tex]
Plugging in the values:
[tex]\[ \text{Lower bound} = \sqrt{ \frac{(109-1) \cdot (0.63)^2}{146.567} } = \sqrt{ \frac{108 \cdot 0.3969}{146.567} } = \sqrt{ \frac{42.0642}{146.567} } \approx 0.54 \][/tex]
[tex]\[ \text{Upper bound} = \sqrt{ \frac{(109-1) \cdot (0.63)^2}{74.222} } = \sqrt{ \frac{108 \cdot 0.3969}{74.222} } = \sqrt{ \frac{42.0642}{74.222} } \approx 0.75 \][/tex]
6. State the final confidence interval:
Therefore, the 98% confidence interval estimate for the standard deviation of body temperature of all healthy humans is:
[tex]\[ 0.54^\circ \text{F} < \sigma < 0.75^\circ \text{F} \][/tex]
Make sure to round your final answers to two decimal places as needed.
