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Answer:
The population standard deviation is not known.
90% Confidence interval by T₁₀-distribution: (38.3, 53.7).
Step-by-step explanation:
The "standard deviation" of $14 comes from a survey. In other words, the true population standard deviation is not known, and the $14 here is an estimate. Thus, find the confidence interval with the Student t-distribution. The sample size is 11. The degree of freedom is thus .
Start by finding 1/2 the width of this confidence interval. The confidence level of this interval is 90%. In other words, the area under the bell curve within this interval is 0.90. However, this curve is symmetric. As a result,
- The area to the left of the lower end of the interval shall be
.
- The area to the left of the upper end of the interval shall be
.
Look up the t-score of the upper end on an inverse t-table. Focus on the entry with
- a degree of freedom of 10, and
- a cumulative probability of 0.95.
.
This value can also be found with technology.
The formula for 1/2 the width of a confidence interval where standard deviation is unknown (only an estimate) is:
,
where
is the t-score at the upper end of the interval,
is the unbiased estimate for the standard deviation, and
is the sample size.
For this confidence interval:
,
, and
.
Hence the width of the 90% confidence interval is
.
The confidence interval is centered at the unbiased estimate of the population mean. The 90% confidence interval will be approximately:
.