# The speeds of vehicles traveling on a highway are normally distributed with an unknown population...

## Question:

The speeds of vehicles traveling on a highway are normally distributed with an unknown population mean and standard deviation. A random sample of 11 vehicles is taken and results in a sample mean of 65 miles per hour and sample standard deviation of 9 miles per hour. Find a 90% confidence interval estimate for the population mean using the Student's t-distribution.

## Confidence Interval

An estimate of the population parameter given by two numbers between which the parameter may be supposed to lie is called interval estimate or confidence interval. A confidence interval has an upper limit and a lower limit.

It is given that,

{eq}\begin{align*} \overline x &= 65\\ n &= 11\\ s &= 9 \end{align*} {/eq}

The degree of freedom

{eq}\begin{align*} \nu &= n - 1\\ &= 11 - 1\\ &= 10 \end{align*} {/eq}

The critical value of t for degrees of freedom 10 and at 10% confidence level is 1.812

The confidence interval is given by the formula,

{eq}\begin{align*} & \overline x \pm t\dfrac{s}{{\sqrt n }}\\ & = 65 \pm 1.812 \times \dfrac{9}{{\sqrt {11} }}\\ & = (65 - 4.918,65 + 4.918)\\ &= (60.082,69.918) \end{align*} {/eq}

Hence, the 90% confidence interval is (60.082, 69.918).