Question

An automobile manufacturer claims that their van has a 46.4 miles/gallon (MPG) rating. An independent testing firm has been contracted to test the MPG for this van. After testing 140 vans they found a mean MPG of 46.0. Assume the standard deviation is known to be 2.6. Is there sufficient evidence at the 0.02 level that the vans underperform the manufacturer's MPG rating?

Answer 1

The decision rule is

Fail to reject the null hypothesis

The conclusion

There is no sufficient evidence to conclude that the vans underperform the manufacturer's MPG rating

Explanation:

From the question we are told that

The population mean is
`\mu = 46.4 \ miles/gallon`

The sample size is n = 140

The sample mean is
`\= x = 46.0`

The standard deviation is
`\sigma = 2.6`

The level of significance is
`\alpha = 0.02`

The null hypothesis is
`H_o : \mu = 46.4`

The alternative hypothesis is
`H_a : \mu < 46.4`

Generally the test statistics is mathematically represented as

`t = ( \= x - \mu )/( (\sigma )/(√(n) ) )`

=>
`t = ( 46 - 46.4 )/( ( 2.6 )/(√(140) ) )`

=>
`t = -1.8203`

From the z table the area under the normal curve to the left corresponding to

-1.8203 is

`P(Z < -1.820 ) = 0.03438`

Generally the p-value is mathematically represented as

`p-value = P(Z < -1.820 ) = 0.03438`

From the value obtained we see that the p-value >
`\alpha` hence

The decision rule is

Fail to reject the null hypothesis

The conclusion

There is no sufficient evidence to conclude that the vans underperform the manufacturer's MPG rating