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A researcher would like to investigate whether or not there is a difference in I...

Apr 4, 2024

A researcher would like to investigate whether or not there is a difference in IQ between Psychology and History majors. She gathers 16 students (8 whose major is Psychology, and 8 whose major is history) and has them all take an IQ test to test this hypothesis using an alpha level of .01. Below are the data: \begin{tabular}{|l|l|} \hline Psych & History \\ \hline 120 & 95 \\ \hline 114 & 92 \\ \hline 100 & 145 \\ \hline 121 & 101 \\ \hline 124 & 100 \\ \hline 102 & 98 \\ \hline 110 & 97 \\ \hline 123 & 120 \\ \hline \end{tabular} What is the appropriate test?

Solution by Steps

step 1

Identify the type of data and the hypothesis test to use: The data consists of two independent samples (Psychology and History majors), and we want to compare their means. Since the sample sizes are small (n < 30) and we do not know the population standard deviations, we use a two-sample t-test assuming equal variances

step 2

State the null hypothesis (

H_0

) and the alternative hypothesis (

H_a

H_0: \mu_{Psych} = \mu_{Hist}

(no difference in means) versus

H_a: \mu_{Psych} \neq \mu_{Hist}

(a difference in means)

step 3

Calculate the sample means (

\bar{x}_{Psych}

and

\bar{x}_{Hist}

) and sample standard deviations (s_{Psych} and s_{Hist}) for each group

step 4

Calculate the test statistic using the formula for the two-sample t-test:

t = \frac{\bar{x}_{Psych} - \bar{x}_{Hist}}{\sqrt{\frac{s_{Psych}^2}{n_{Psych}} + \frac{s_{Hist}^2}{n_{Hist}}}}

where

n_{Psych}

and

n_{Hist}

are the sample sizes for the Psychology and History groups, respectively

step 5

Determine the degrees of freedom (df) for the test. Since we are assuming equal variances, we can use the formula:

df = n_{Psych} + n_{Hist} - 2

step 6

Find the critical t-value for a two-tailed test at the

\alpha = .01

level using the t-distribution table and the calculated degrees of freedom

step 7

Compare the calculated t-statistic to the critical t-value to determine whether to reject or fail to reject the null hypothesis

Answer

The appropriate test is the two-sample t-test assuming equal variances.

Key Concept

Two-sample t-test for independent samples

Explanation

The two-sample t-test is used to determine if there is a statistically significant difference between the means of two independent samples when the population standard deviations are unknown and the sample sizes are small.