A random sample of 16 ATM transactions at the Last National Bank of Flat Rock revealed a mean transaction time of 2.8 minutes with a standard deviation of 1.2 minutes. The width (in minutes) of the 95 percent confidence interval for the true mean transaction time is:
A. ± 0.639
B. ± 0.588
C. ± 0.300
D. ± 2.131
The correct answer and explanation is :
To determine the width of a 95% confidence interval for the true mean transaction time, we use the t-distribution because the sample size is small (n < 30) and the population standard deviation is unknown.
Step 1: Given Data
- Sample size ( n = 16 )
- Sample mean ( \bar{x} = 2.8 ) minutes (not directly needed for width)
- Sample standard deviation ( s = 1.2 ) minutes
- Confidence level = 95%
Step 2: Determine the t-score
Since the sample size is 16, degrees of freedom (df) = ( n – 1 = 15 ).
From the t-distribution table, the critical value for a 95% confidence interval and 15 degrees of freedom is approximately:
[
t_{0.025, 15} = 2.131
]
Step 3: Compute the Standard Error (SE)
[
SE = \frac{s}{\sqrt{n}} = \frac{1.2}{\sqrt{16}} = \frac{1.2}{4} = 0.3
]
Step 4: Compute the Margin of Error (ME)
[
ME = t \times SE = 2.131 \times 0.3 = 0.6393 \approx 0.639
]
So the width of the confidence interval is:
[
\pm 0.639 \text{ minutes}
]
Final Answer:
A. ± 0.639
Explanation:
A confidence interval provides a range of values within which we believe the true population mean lies, with a certain level of confidence (95% in this case). The formula for the confidence interval when the population standard deviation is unknown uses the t-distribution, which adjusts for sample variability.
The width of this interval is determined by multiplying the standard error by the appropriate t-critical value, which depends on both the confidence level and the degrees of freedom. A smaller sample size results in a larger t-value, reflecting increased uncertainty.
The result ( \pm 0.639 ) means we are 95% confident that the true mean ATM transaction time lies within 0.639 minutes above or below the sample mean of 2.8 minutes.