A sample of 106 body temperatures resulted in a mean of 98.2 degrees and a standard deviation of .62 degrees. Use these sample statistics to construct a 98% confidence interval estimate of the standard deviation of body temperature of all healthy humans.
To construct a confidence interval for the standard deviation, we can use the chi-square distribution. The formula for the confidence interval is:
( n - 1 ) s² / χ²(L/2) ≤ σ² ≤ ( n - 1 ) s² / χ²(1-L/2)
Where:
n = sample size
s = sample standard deviation
χ²(L/2) = chi-square value at the lower critical point (L/2) with (n-1) degrees of freedom
χ²(1-L/2) = chi-square value at the upper critical point (1-L/2) with (n-1) degrees of freedom
L = level of confidence (in decimal form)
Let's calculate the confidence interval step-by-step:
Step 1: Identify the values
- Sample size (n) = 106
- Sample standard deviation (s) = 0.62
- Level of confidence (L) = 0.98 (because it's a 98% confidence interval)
Step 2: Find the chi-square values
Using a chi-square distribution table or calculator, find the chi-square values at the lower and upper critical points corresponding to a 98% confidence level with (n-1) degrees of freedom.
χ²(L/2) = χ²(0.01/2) = χ²(0.005) = 143.4096
χ²(1-L/2) = χ²(1-0.01/2) = χ²(0.995) = 76.7939
Step 3: Calculate the confidence interval
Plug in the values into the formula:
Lower bound: (n - 1) * s² / χ²(L/2) = 105 * 0.62² / 143.4096 ≈ 0.2602125
Upper bound: (n - 1) * s² / χ²(1-L/2) = 105 * 0.62² / 76.7939 ≈ 0.4200584
The 98% confidence interval estimate of the standard deviation of body temperature of all healthy humans is approximately 0.2602125 to 0.4200584 degrees.
To construct a confidence interval estimate for the standard deviation of body temperature, we can use the Chi-Square distribution.
The formula for calculating the confidence interval estimate for the standard deviation is:
CI = [(n - 1) * s^2 / X^2, (n - 1) * s^2 / X^2]
Where:
- n is the sample size (106 in this case)
- s is the sample standard deviation (.62 in this case)
- X^2 is the critical value from the Chi-Square distribution
To find the critical value, we need to determine the degrees of freedom. Since we are estimating the standard deviation, the degrees of freedom is given by (n - 1).
For a 98% confidence level, we want a two-tailed test, so our significance level is (1 - confidence level) / 2, which is (1 - 0.98) / 2 = 0.01. Looking up the critical value for 0.01 and (n - 1) degrees of freedom, we can find the value from the Chi-Square distribution table.
Let's assume the critical value is denoted as X^2. Plugging in the given values, the confidence interval estimate can be calculated as:
CI = [(n - 1) * s^2 / X^2, (n - 1) * s^2 / X^2]
= [(106 - 1) * 0.62^2 / X^2, (106 - 1) * 0.62^2 / X^2]
Now, it's time to look up the critical value from the Chi-Square distribution table with 105 degrees of freedom and a significance level of 0.01. The critical value we find will be used instead of X^2 in the formula above.
Once you find the critical value, substitute it in the formula, evaluate the expression, and calculate the confidence interval estimate for the standard deviation.
98% = mean ± Z SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.01) and its Z score. Insert data and calculate.