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\( 1 \leftarrow \begin{array}{l}\text { Researchers conducted an experiment to test the effects of alcohol. Erors were recorded in a test of visual and motor skills for a treatment group of } 29 \text { people who drank ethanol and another group of } 29 \text { people given a } \\ \text { placebo. The errors for the treatment group have a standard deviation of } 230 \text {, and the errors for the placebo group have a standard deviation of } 0.87 \text {. Assume that the two populations are normally distributed. Use a } 0.05 \\ \text { significance level to test the claim that both groups have the same amount of variation among the errors. }\end{array} \)

Ask by Kelley Estrada. in the United States
Mar 11,2025

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There is significant evidence to suggest that the treatment and placebo groups have different amounts of variation in their errors.

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To tackle the question at hand, we first need to conduct an F-test for equality of variances. We’ll start by calculating the F-ratio, which is the ratio of the two sample variances. The formula is: \[ F = \frac{s_1^2}{s_2^2} \] where \(s_1^2\) is the variance of the treatment group and \(s_2^2\) is the variance of the placebo group. With the standard deviations given: \[ s_1 = 230 \quad \Rightarrow \quad s_1^2 = 230^2 = 52900 \] \[ s_2 = 0.87 \quad \Rightarrow \quad s_2^2 = 0.87^2 \approx 0.7569 \] Then we can calculate the F-ratio: \[ F = \frac{52900}{0.7569} \approx 69860.82 \] Next, we find the critical value of F at a significance level of 0.05 with degrees of freedom \(df_1 = n_1 - 1 = 29 - 1 = 28\) for the treatment group and \(df_2 = n_2 - 1 = 29 - 1 = 28\) for the placebo group. This statistic can be looked up in an F-distribution table or calculated using statistical software. The result indicates the extreme difference in variances, leading to the conclusion that we reject the null hypothesis, which claims that both groups have the same amount of variation. This analysis shows that the treatment group exhibits significantly greater variability in errors compared to the placebo group at the α = 0.05 significance level. In conclusion, the implication is that consuming ethanol has a substantial effect on visual and motor skills, reflected in the wide range of error variation.

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