Question

A college student is interested in investigating the TV-watching habits of her classmates and surveys 20 people on the number of hours they watch per week. The results are provided below. Calculate the 80% confidence interval of the true average number of hours of TV watched per week.

P.S: excel formola needed only. For lower and Upper Bound
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Answer 1

80% confidence interval of the true average number of hours of TV watched per week is [8.28 hours, 11.02 hours].

Explanation:

We are given that a college student is interested in investigating the TV-watching habits of her classmates and surveys 20 people on the number of hours they watch per week. The results are provided below;

Hours of TV per week (X): 6, 14, 13, 6, 16, 10, 19, 4, 5, 5, 18, 8, 7, 14, 8, 8, 9, 12, 6, 5.

Firstly, the Pivotal quantity for 80% confidence interval for the true average is given by;

P.Q. =
`(\bar X-\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean number of hours of TV watched per week =
`(\sum X)/(n)` = 9.65

s = sample standard deviation =
`\sqrt{(\sum (X -\bar X)^(2) )/(n-1) }` = 4.61

n = sample of people = 20

`\mu` = true average number of hours of TV watched per week

Here for constructing 80% confidence interval we have used One-sample t-test statistics as we don't know about population standard deviation.

So, 80% confidence interval for the true average,
`\mu` is ;

P(-1.33 <
`t_1_9` < 1.33) = 0.80 {As the critical value of t at 19 degrees of

freedom are -1.33 & 1.33 with P = 10%}

P(-1.33 <
`(\bar X-\mu)/((s)/(√(n) ) )` < 1.33) = 0.80

P(
`-1.33 * {(s)/(√(n) ) }` <
`{\bar X-\mu}` <
`1.33 * {(s)/(√(n) ) }` ) = 0.80

P(
`\bar X-1.33 * {(s)/(√(n) ) }` <
`\mu` <
`\bar X+1.33 * {(s)/(√(n) ) }` ) = 0.80

80% confidence interval for
`\mu` = [
`\bar X-1.33 * {(s)/(√(n) ) }` ,
`\bar X+1.33 * {(s)/(√(n) ) }` ]

= [
`9.65-1.33 * {(4.61)/(√(20) ) }` ,
`9.65+1.33 * {(4.61)/(√(20) ) }` ]

= [8.28 hours, 11.02 hours]

Therefore, 80% confidence interval of the true average number of hours of TV watched per week is [8.28 hours, 11.02 hours].