Question

A substance abuse researcher wants to know whether reaction time is decreased by frequent alcohol use. He randomly selects 164 alcoholics and finds that their mean reaction time (in ms) to a stimulus equals 283. Reaction times to the stimulus in the general population are distributed normally with a mean equal to 281 and a standard deviation equal to 28. Is there sufficient evidence at the 0.01 significance level to conclude that reaction time is decreased by frequent alcohol use

Answer 1

We conclude that the reaction time is increased or remains same by frequent alcohol use.

Explanation:

We are given that a substance abuse researcher wants to know whether reaction time is decreased by frequent alcohol use. He randomly selects 164 alcoholics and finds that their mean reaction time (in ms) to a stimulus equals 283.

Reaction times to the stimulus in the general population are distributed normally with a mean equal to 281 and a standard deviation equal to 28.

Let
`\mu` = mean reaction time after alcohol use.

So, Null Hypothesis,
`H_0` :
`\mu`
`\geq` 281 {means that the reaction time is increased or remains same by frequent alcohol use}

Alternate Hypothesis,
`H_A` :
`\mu` < 281 {means that the reaction time is decreased by frequent alcohol use}

The test statistics that would be used here One-sample z test statistics as we know about population standard deviation;

T.S. =
`(\bar X-\mu)/((\sigma)/(√(n)))` ~ N(0,1)

where,
`\bar X` = sample mean reaction time (in ms) = 283

`\sigma` = population standard deviation = 28

n = sample of alcoholics = 164

So, test statistics =
`(283-281)/((28)/(√(164)))`

= 0.915

The value of z test statistics is 0.915.

Now, at 0.01 significance level the z table gives critical value of -2.3263 for left-tailed test. Since our test statistics is more than the critical values of z, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.

Therefore, we conclude that the reaction time is increased or remains same by frequent alcohol use.