Question

The Highway Safety Department wants to study the driving habits of individuals. A sample of 37 cars traveling on a particular stretch of highway revealed an average speed of 70.7 miles per hour with a standard deviation of 6.3 miles per hour. Round to 4 decimal places. 1.Calculate a 90% confidence interval for the true mean speed of all cars on this particular stretch of highway

Answer 1

90% confidence interval for the true mean speed of all cars on this particular stretch of highway is [68.9517 miles per hour , 72.4483 miles per hour].

Explanation:

We are given that a sample of 37 cars traveling on a particular stretch of highway revealed an average speed of 70.7 miles per hour with a standard deviation of 6.3 miles per hour.

Firstly, the pivotal quantity for 90% confidence interval for the true mean is given by;

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

where,
`\bar X` = sample average speed of cars = 70.7 miles per hour

s = sample standard deviation = 6.3 miles per hour

n = sample of cars = 37

`\mu` = true mean speed

Here for constructing 90% confidence interval we have used One-sample t test statistics as we know don't about population standard deviation.

So, 90% confidence interval for the true mean,
`\mu` is ;

P(-1.688 <
`t_3_6` < 1.688) = 0.90 {As the critical value of t at 36 degree of

freedom are -1.688 & 1.688 with P = 5%}

P(-1.688 <
`(\bar X-\mu)/((s)/(√(n) ) )` < 1.688) = 0.90

P(
`-1.688 * {(s)/(√(n) ) }` <
`{\bar X-\mu}` <
`1.688 * {(s)/(√(n) ) }` ) = 0.90

P(
`\bar X-1.688 * {(s)/(√(n) ) }` <
`\mu` <
`\bar X+1.688 * {(s)/(√(n) ) }` ) = 0.90

90% confidence interval for
`\mu` = [
`\bar X-1.688 * {(s)/(√(n) ) }` ,
`\bar X+1.688 * {(s)/(√(n) ) }` ]

= [
`70.7-1.688 * {(6.3)/(√(37) ) }` ,
`70.7+1.688 * {(6.3)/(√(37) ) }` ]

= [68.9517 miles per hour , 72.4483 miles per hour]

Therefore, 90% confidence interval for the true mean speed of all cars on this particular stretch of highway is [68.9517 miles per hour , 72.4483 miles per hour].

The interpretation of the above interval is that we are 90% confident that the true mean speed of all cars will lie between 68.9517 miles per hour and 72.4483 miles per hour.