Math  /  Data & Statistics

QuestionA teaching assistant collected data from students in one of her classes to investigate whether study time per week (average number of hours) differed between students in the class who planned to go to graduate school and those who did not. Complete parts (a) through (c).
Click the icon to view the data. s1=8.22s_{1}=8.22 (Round to the nearest hundredth as needed.) Find the standard deviation for students who did not plan to go to graduate school. s2=3.35s_{2}=3.35 (Round to the nearest hundredth as needed.) Interpret these values. A. The sample mean was higher for the students who planned to go to graduate school, but the tir B. The sample mean was lower for the students who planned to go to graduate school, but the tim C. The sample mean was lower for the students who planned to go to graduate school, but the tim D. The sample mean was higher for the students who planned to go to graduate school, but the tir Print
Data table \begin{tabular}{|c|} \hline Fraduate school: 17,7,15,10,5,5,2,3,12,16,15,35,817,7,15,10,5,5,2,3,12,16,15,35,8, \\ 14,10,19,3,26,15,5,514,10,19,3,26,15,5,5 \end{tabular} b. Find the standard error for the difference between the sample means. Interpret.
Find the standard error for the difference between the sample means. se=2.08s e=2.08 (Round to the nearest hundredth as needed.) Interpret this value. A. If further random samples of these sizes were obtained from these populations, the differences between the sample means would vary. The standard deviation of these values for ( xˉ1xˉ2)\left.\bar{x}_{1}-\bar{x}_{2}\right) would equal about 2.1. B. If further random samples of these sizes were obtained from these populations, the differences between the sample means would not vary. The value of ( xˉ1xˉ2)\left.\bar{x}_{1}-\bar{x}_{2}\right) would equal about 2.1 . C. If further random samples of these sizes were obtained from these populations, the differences between the sample means would vary. The standard deviation of these values for ( xˉ1xˉ2)\left.\bar{x}_{1}-\bar{x}_{2}\right) would equal about 4.2 .

Studdy Solution

STEP 1

What is this asking? We're comparing the study habits of students who plan to go to graduate school versus those who don't, using standard deviations and standard error. Watch out! Don't mix up standard deviation (spread within a group) and standard error (spread of the difference between groups)!

STEP 2

1. Calculate the standard deviation for graduate school.
2. Interpret the standard deviation.

STEP 3

The hours studied by students planning to go to graduate school are: 17, 7, 15, 10, 5, 5, 2, 3, 12, 16, 15, 35, 8, 14, 10, 19, 3, 26, 15, 5, 5.

STEP 4

**Calculate the sum:** 17+7+15+10+5+5+2+3+12+16+15+35+8+14+10+19+3+26+15+5+5=25717 + 7 + 15 + 10 + 5 + 5 + 2 + 3 + 12 + 16 + 15 + 35 + 8 + 14 + 10 + 19 + 3 + 26 + 15 + 5 + 5 = 257. **Divide by the number of data points:** There are **21** data points.
So, the **mean** is 2572112.24\frac{257}{21} \approx 12.24.

STEP 5

For each data point, subtract the mean and square the result.
For example, the first data point (17) gives us (1712.24)222.13(17 - 12.24)^2 \approx 22.13.
Doing this for all data points, we get approximately: 22.13, 27.46, 7.57, 4.93, 52.42, 52.42, 104.98, 85.43, 0.06, 14.14, 7.57, 509.35, 17.98, 3.1, 4.93, 45.62, 85.43, 187.42, 7.57, 52.42, 52.42.

STEP 6

Adding all those squared differences, we get approximately **1282.94**.

STEP 7

Divide the sum of squared differences by one less than the number of data points (211=2021-1=20): 1282.942064.15\frac{1282.94}{20} \approx 64.15.
This is the **variance**.

STEP 8

The **standard deviation** is the square root of the variance: 64.158.01\sqrt{64.15} \approx 8.01.

STEP 9

The standard deviation tells us how spread out the data is.
A larger standard deviation means the data is more spread out, while a smaller standard deviation means the data is more clustered around the mean.

STEP 10

In this case, the standard deviation of approximately **8.01 hours** tells us that the weekly study times for students planning to go to graduate school are quite varied.
Some students study much more or much less than the average of roughly **12.24 hours**.

STEP 11

The standard deviation for students planning to go to graduate school is approximately **8.01 hours**.
This indicates a significant variation in study time among these students.

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