Math  /  Data & Statistics

QuestionGUIDED SOLUTION
Essays A professor wishes to see if two groups of students' essays differ in lengths, that is, the number of words in each essay. The professor randomly selects 12 essays from a group of students who are science majors and 10 essays from a group of humanities majors to compare. The data are shown. At α=0.10\alpha=0.10, can it be concluded that there is a difference in the lengths of the essays between the two groups?
Science majors  Science majors 205731292501346427622075298437833259238939393425\begin{array}{llllllllll} \text { Science majors } \\ \hline 2057 & 3129 & 2501 & 3464 & 2762 & 2075 & 2984 & 3783 & 3259 & 2389 \\ 3939 & 3425 \end{array}
Humanities majors  Humanities majors 269829402257193321740\begin{array}{l} \text { Humanities majors } \\ \hline 2698 \\ \hline 2940 \\ 2257 \\ 19332 \\ 1740 \end{array}
Send data to Excel Use μ1\mu_{1} for the mean of science majors and μ2\mu_{2} for the mean of humanities majors. Assume the populations are normally distributed, and that the variances are unequal. (a) State the hypotheses and identify the claim with the correct hypothesis. (b) Find the critical value(s). (c) Compute the test value. (d) Make the decision. (e) Summarize the results. (a) State the hypotheses and identify the claim with the correct hypothesis.
The null hypothesis H0H_{0} is the statement that there is (Choose one) \nabla between the means. This is equivalent to μ1\mu_{1} (Choose one) μ2\nabla \mu_{2}.

Studdy Solution

STEP 1

What is this asking? We want to see if there's a real difference in essay lengths between science and humanities majors, using a sample of their essays. Watch out! We're told the variances are unequal, so we'll need to use the correct t-test formula!
Also, don't mix up the groups when calculating the means.

STEP 2

1. State the Hypotheses
2. Find the Critical Value
3. Compute the Test Value
4. Make the Decision
5. Summarize the Results

STEP 3

Our *null hypothesis* (H0H_0) is that there's **no difference** in the average essay length between science and humanities majors.
Mathematically, this means μ1=μ2\mu_1 = \mu_2, where μ1\mu_1 is the average essay length for science majors and μ2\mu_2 is the average essay length for humanities majors.

STEP 4

The *alternative hypothesis* (H1H_1) is that there *is* a difference in average essay lengths.
This means μ1μ2\mu_1 \ne \mu_2.
This is a *two-tailed test* since we're looking for any difference, not specifically whether one group writes longer essays.

STEP 5

Since we have unequal variances, we use a slightly complicated formula for *degrees of freedom*: df=(s12n1+s22n2)2(s12n1)2n11+(s22n2)2n21\text{df} = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} Where s1s_1 and s2s_2 are the **sample standard deviations**, and n1n_1 and n2n_2 are the **sample sizes**.

STEP 6

We're given α=0.10\alpha = 0.10, and it's a two-tailed test.
We'll look up the critical t-value in a t-table (or use a calculator) once we have the degrees of freedom.

STEP 7

First, we **calculate the mean** (xˉ\bar{x}) and **standard deviation** (ss) for each group.
*Science Majors:* xˉ1=2993.58\bar{x}_1 = 2993.58, s1=631.95s_1 = 631.95
*Humanities Majors:* xˉ2=2653.6\bar{x}_2 = 2653.6, s2=530.58s_2 = 530.58

STEP 8

Now, we use the following formula for the *t-test statistic*: t=xˉ1xˉ2s12n1+s22n2=2993.582653.6631.95212+530.582101.17t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} = \frac{2993.58 - 2653.6}{\sqrt{\frac{631.95^2}{12} + \frac{530.58^2}{10}}} \approx 1.17

STEP 9

Plugging in the values for s1s_1, s2s_2, n1=12n_1 = 12, and n2=10n_2 = 10 into the degrees of freedom formula, we get: df19.96\text{df} \approx 19.96 We can round this down to **19** degrees of freedom.

STEP 10

With 19 degrees of freedom and α=0.10\alpha = 0.10 for a two-tailed test, our critical t-value is approximately ±1.729\pm 1.729.

STEP 11

Our calculated test value (t1.17t \approx 1.17) falls *within* the critical region (1.729<1.17<1.729-1.729 < 1.17 < 1.729).
Therefore, we **fail to reject the null hypothesis**.

STEP 12

There's not enough evidence to conclude that there's a significant difference in essay lengths between science and humanities majors at the α=0.10\alpha = 0.10 significance level.

STEP 13

There is no significant difference in the lengths of the essays between the two groups at the 0.10 significance level.

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