Question

Paint used to paint lines on roads must reflect enough light to be clearly visible at night. Let μ denote the true average reflectometer reading for a new type of paint under consideration. A test of H0: μ = 20 versus Ha: μ > 20 based on a sample of 13 observations gave t = 3.0.

What conclusion is appropriate at each of the following significance levels?

(a) α = 0.05 Reject H0 Fail to reject H0
(b) α = 0.01 Reject H0 Fail to reject H0
(c) α = 0.001 Reject H0 Fail to reject H0

Answer 1

`p_v =P(t_(12)>3.0)=0.0055`

(c) α = 0.001 Reject H0 Fail to reject H0

TRUE, Since the p value is higher than the significance level
`p_v >\alpha` we have enough evidence to FAIL to reject the null hypothesis. And we conclude that the true mean is NOT significantly higher than 20.

Explanation:

Previous concepts and data given

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X` represent the sample mean

`s` represent the sample standard deviation

n=13 represent the sample selected

`\alpha` significance level

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the true mean is higher than 20, the system of hypothesis would be:

Null hypothesis:
`\mu \leq 20`

Alternative hypothesis:
`\mu > 20`

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic

On this case we have the t statisitc calculated and is
`t_(calc)=3.0`

P-value

First we need to calculate the degrees of freedom given by:

`df=n-1=13-1 =12`

Then since is a right tailed test the p value would be:

`p_v =P(t_(12)>3.0)=0.0055`

Conclusion

(a) α = 0.05 Reject H0 Fail to reject H0

FALSE, Since the p value is less than the significance level
`p_v <\alpha` we have enough evidence to reject the null hypothesis. And we conclude that the true mean is significantly higher than 20.

(b) α = 0.01 Reject H0 Fail to reject H0

FALSE, Since the p value is less than the significance level
`p_v <\alpha` we have enough evidence to reject the null hypothesis. And we conclude that the true mean is significantly higher than 20.

(c) α = 0.001 Reject H0 Fail to reject H0

TRUE, Since the p value is higher than the significance level
`p_v >\alpha` we have enough evidence to FAIL to reject the null hypothesis. And we conclude that the true mean is NOT significantly higher than 20.