Question

An automobile manufacturer claims that its van has a 31.3 miles/gallon (MPG) rating. An independent testing firm has been contracted to test the MPG for this van since it is believed that the van has an incorrect manufacturer's MPG rating. After testing 140 vans, they found a mean MPG of 31.1. Assume the standard deviation is known to be 1.3. A level of significance of 0.02 will be used. Make a decision to reject or fail to reject the null hypothesis.

Answer 1

We conclude that there is enough evidence to claim that the van has a 31.3 miles/gallon (MPG) rating.

Explanation:

We are given the following in the question:

Population mean, μ = 31.3 miles/gallon

Sample mean,
`\bar{x}` = 31.1

Sample size, n = 140

Alpha, α = 0.02

Population standard deviation, σ = 1.3

First, we design the null and the alternate hypothesis

`H_(0): \mu = 31.3\text{ miles/gallon}\\H_A: \mu \\eq 31.3\text{ miles/gallon}`

We use Two-tailed z test to perform this hypothesis.

Formula:

`z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`z_(stat) = \displaystyle(31.1 - 31.3)/((1.3)/(√(140)) ) = -1.82`

Now,
`z_(critical) \text{ at 0.02 level of significance } = \pm 2.33`

Since,

The calculated z-statistic lies in the acceptance region, we fail to reject the null hypothesis and accept it.

We conclude that there is enough evidence to claim that the van has a 31.3 miles/gallon (MPG) rating.