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Answer :
The 95% confidence interval for the mean body temperature is approximately: (97.038, 98.646)
To find the 95% confidence interval for the mean body temperature, we can use the formula for the confidence interval for a population mean when the population is normally distributed and the population standard deviation is unknown:
Confidence Interval Formula:
CI = x ± t * (s / √n)
Where:
x is the sample mean
s is the sample standard deviation
n is the sample size
t is the t-critical value for a 95% confidence level with n - 1 degrees of freedom
Step 1: Calculate the sample mean
96.9, 99.7, 98.4, 97.5, 96.8, 97.4, 96.6, 98.7, 99.6, 99.8, 96.5, 97.2
Sum of the values = 96.9 + 99.7 + 98.4 + 97.5 + 96.8 + 97.4 + 96.6 + 98.7 + 99.6 + 99.8 + 96.5 + 97.2
= 1174.1
Sample mean = Sum / n = 1174.1 / 12
= 97.8417
Step 2: Calculate the sample standard deviation (s)
The sample variance formula is:
s² = Σ(xi - x)² / (n - 1)
First, subtract the mean from each data point, square the result, and sum them up:
(96.9 - 97.8417)² = 0.8892 (99.7 - 97.8417)² = 3.4482 (98.4 - 97.8417)² = 0.3104 (97.5 - 97.8417)² = 0.1160 (96.8 - 97.8417)² = 1.0877 (97.4 - 97.8417)² = 0.1922 (96.6 - 97.8417)² = 1.5524 (98.7 - 97.8417)² = 0.7399 (99.6 - 97.8417)² = 3.0761 (99.8 - 97.8417)² = 3.8292 (96.5 - 97.8417)² = 1.7831 (97.2 - 97.8417)²
= 0.4062
Sum of squared differences = 17.5183
Sample variance (s²) = 17.5183 / (12 - 1) = 1.5962
Sample standard deviation (s) = √1.5962 = 1.264
Step 3: Find the t-critical value for a 95% confidence interval
Since we have 12 data points, the degrees of freedom (df) is 12 - 1 = 11. Using a t-table or calculator for a 95% confidence level and df = 11, the t-critical value (t) is approximately 2.201.
Step 4: Calculate the margin of error
Margin of error = t * (s / √n)
Margin of error = 2.201 * (1.264 / √12) = 2.201 * (1.264 / 3.464)
= 2.201 * 0.365
= 0.804
Step 5: Calculate the confidence interval
Lower bound = x - margin of error = 97.8417 - 0.804
= 97.0377
Upper bound = x + margin of error = 97.8417 + 0.804
= 98.6457
Complete question
The body temperatures (in degrees Fahrenheit) of a sample of adults from a small town are as follows:
96.9, 99.7, 98.4, 97.5, 96.8, 97.4, 96.6, 98.7, 99.6, 99.8, 96.5, 97.2
Assume that the body temperatures of adults in this town are normally distributed. Based on this data, calculate the 95% confidence interval for the mean body temperature of adults in the town.
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