A researcher is interested in finding a 90% confidence interval for the mean number minutes students are concentrating on their professor during a one

Question

A researcher is interested in finding a

90 %

confidence interval for the mean number minutes students are concentrating on their professor during a one hour statistics lecture. The study included 117 students who averaged 40.9 minutes concentrating on their professor during the hour lecture. The standard deviation was 11.8 minutes. Round answers to 3 decimal places where possible.
a.
To compute the confidence interval use a ? distribution.
b.
With

90 %

confidence the population mean minutes of concentration is between ____ and ____ minutes.
c.
If many groups of 117 randomly selected students are studied, then a different confidence interval would be produced from each group. About ____ percent of these confidence intervals will contain the true population mean minutes of concentration and about ____ percent will not contain the true population mean number of minutes of concentration.

Answer 1

Step 1
Given: $\overset{―}{x} = 40.9, s = 11.8, n = 117$
a)
The sampling distribution follows a t-distribution.
Reason:
Since the population standard deviation is unknown, so the t-distribution is appropriate
b)The degree of freedom is,
$d f = n - 1$
$= 117 - 1$
$= 116$
Critical value:
By using the t-tables, the critical value at $10 %$ level of significance for a two tailed t -distribution is,
$t_{0.10 / 2.116} = \pm 1.658$
Step 2
The $90 %$ confidence interval for the mean number minutes students are concentrating on their professor during a one hour statistics lecture is,
$90 % C I = \overset{―}{x} - t_{α / 2} (\frac{s}{\sqrt{n}}) < μ < \overset{―}{x} + t_{α / 2} (\frac{s}{\sqrt{n}})$
$= 40.9 - 1.658 (\frac{11.8}{\sqrt{117}}) < μ < 40.9 + 1.658 (\frac{11.8}{\sqrt{117}})$
$= 40.9 - 1.658 (1.090910386) < μ < 40.9 + 1.658 (1.090910386)$
$= 40.9 - 1.80872942 < μ < 40.9 + 1.80872942$
$= 39.09127058 < μ < 42.70872942$
$= 39.091 < μ < 42.709$
With $95 %$ confidence the population the mean minutes of concentration is between 39.091 and 42.709.
c)
If many groups of 117 randomly selected students are studied, then a different confidence interval would be produced from each group. About 95 percent of these confidence intervals will contain the true population mean number of minutes of concentration and about 5 percent will not contain the true population mean number of minutes of concentration.

A researcher is interested in finding a 90% confidence interval for the mean number minutes students are concentrating on their professor during a one

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