Testing Effects of Alcohol Researchers conducted an experiment...

Question

Testing Effects of Alcohol Researchers conducted an experiment to test the effects of alcohol. Errors were recorded in a test of visual and motor skills for a treatment group of 22 people who drank ethanol and another group of 22 people given a placebo. The errors for the treatment group have a standard deviation of 2.20, and the errors for the placebo group have a standard deviation of 0.72 (based on data from “Effects of Alcohol Intoxication on Risk Taking, Strategy, and Error Rate in Visuomotor Performance,” by Streufert et al., Journal of Applied Psychology, Vol. 77, No. 4). Use a 0.05 significance level to test the claim that both groups have the same amount of variation among the errors.

Accepted Answer

Hello everyone. Let's take a look at this question together. A psychologist wants to compare the variability in reaction times between two groups those who slept 8 hours with a sample size of 20 and a standard deviation of 0.95 seconds, and those who slept 4 hours with a sample size of 20 and a standard deviation of 0.38 seconds. At the 0.05 significance level, test the claim that both groups have the same amount of variation in reaction times. So in order to solve this question, we must first do our requirement check, which includes that the two populations are independent as the data collected from one group does not influence or relate to the data from the other group, as well as we assume that the two samples are simple random samples and that the psychologist carefully planned the design for the study so that the selection process follows a random procedure. And we assume that each of the two populations is assumed to be normally distributed. And so the first step is to state the hypotheses where we have our null hypothesis that the variance 1 is equal to the variance 2 compared to our alternative hypothesis that the variance 1 is not equal to the variance 2. And so we are testing both of these hypotheses at the alpha equals 0.05 level. And now we can calculate the test statistic F. Where the test statistic F is equal to the variance of group 1 divided by the variance of group 2, where we know that the first variance is the variance of the group with the larger sample standard deviation and the second variance is the variance of the group with the smaller sample standard deviation and based on the information provided in the question, we know. That we have a group that slept 8 hours with a sample size of 20 and a standard deviation of 0.95, as well as a group that slept 4 hours with a sample size of 20 and a standard deviation of 0.38. And since 0.95 is greater than 0.38, and our variance group 1 is the variance of the group with the larger Sample standard deviation, we use the formula for F, which is equal to 0.95 squared divided by 0.38 squared, giving us approximately 6.250 as the value for our test statistic F. And then we determine our degrees of freedom, which our first degree of freedom is equal to N1 minus 1, so 20 minus. 1 is equal to 19, and our second degree of freedom is equal to n2 minus 1, which is also equal to 19. And next we find the critical value which for a two-tail test at alpha is equal to 0.05. We look up the upper critical value for the calculated degrees of freedom and alpha divided by 2 equaling 0.025. And using the table for F distribution with an area of 0.025 in the right tail, we have the following values, and since our first degree of freedom, equaling 19 is between 19 and 20, we need to perform linear interpolation. So, by linear interpolation, where we let the upper critical value be F upper, we have F upper minus 2.6171 divided by 19 minus 15, equaling 2.5089 minus 2.6171, divided by 20 minus 15. And rearranging the formula to solve for F sub upper, we have 2.5089 minus 2.6171 divided by 20 minus 15, all multiplied by 19 minus 15, plus 2.6171, which gives us an F upper value of approximately 2.531. So the upper critical value is 2.531. And since we made a condition that the larger variance is placed in the numerator of the F test statistic, we only need to find the right tailed or upper critical value. And now we can make our decision where if our F test statistic is greater than 2.531, we reject the null hypothesis, and since 6.250 is greater than 2.531, we do in In fact, reject the null hypothesis. So in conclusion, at the 0.05 significance level, there is sufficient evidence to warrant rejection of the claim that both groups have the same amount of variation in reaction times, which makes answer choice A the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

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Key Concepts

Standard Deviation

Hypothesis Testing

Significance Level

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