You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:
| 26.8 | 18.9 | 33.3 | 15.3 | 35.9 | 10 | 21.8 | 8.6 | 23.3 | 36.6 | 16.2 | 6.1 |
Find the [tex]$95 \%$[/tex] confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to 3 decimal places. Assume the data is from a normally distributed population.
Answer (1)
Calculate the sample mean: x ˉ ≈ 21.067 .
Calculate the sample standard deviation: s ≈ 10.501 .
Determine the t-score: t ≈ 2.201 .
Calculate the confidence interval: ( 14.395 , 27.739 ) .
( 14.395 , 27.739 )
Explanation
Problem Analysis We are given a sample of 12 temperatures and asked to find the 95% confidence interval for the mean temperature, assuming a normal distribution.
Calculate Sample Mean First, calculate the sample mean ( x ˉ ) of the temperatures: x ˉ = 12 26.8 + 18.9 + 33.3 + 15.3 + 35.9 + 10 + 21.8 + 8.6 + 23.3 + 36.6 + 16.2 + 6.1 = 12 252.8 = 21.066666666666666
Calculate Sample Standard Deviation Next, calculate the sample standard deviation ( s ) .
s = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2 Using the provided data, the sample standard deviation is approximately 10.501 .
Determine T-Score Determine the t-score ( t α /2 , n − 1 ) for a 95% confidence interval with n − 1 = 12 − 1 = 11 degrees of freedom. For a 95% confidence interval, α = 1 − 0.95 = 0.05 , so α /2 = 0.025 . The t-score for 11 degrees of freedom and α /2 = 0.025 is approximately 2.201 .
Calculate Margin of Error Calculate the margin of error ( E ) :
E = t α /2 , n − 1 ⋅ n s = 2.201 ⋅ 12 10.501 ≈ 6.672
Calculate Confidence Interval Bounds Calculate the lower and upper bounds of the confidence interval: Lower bound: x ˉ − E = 21.067 − 6.672 = 14.395 Upper bound: x ˉ + E = 21.067 + 6.672 = 27.739
State the Confidence Interval Therefore, the 95% confidence interval is ( 14.395 , 27.739 ) .
Examples
Confidence intervals are used in various real-world applications. For example, in medical research, a 95% confidence interval might be calculated for the effectiveness of a new drug. This interval provides a range within which the true effect of the drug is likely to fall. Similarly, in market research, confidence intervals can estimate the range of customer satisfaction scores for a product. These intervals help decision-makers understand the uncertainty associated with sample estimates and make more informed conclusions about the population. The confidence interval provides a plausible range of values for the true population parameter.
