The College Board provided comparisons of Scholastic...

The College Board provided comparisons of Scholastic Aptitude Test (SAT) scores based on the highest level of education attained by the test taker's parents. A research hypothesis was that students whose parents had attained a higher level of education would on average score higher on the SAT. During $2003,$ the overall mean SAT verbal score was 507 (The World Almanac, 2004 ). SAT verbal scores for independent samples of students follow. The first sample shows the SAT verbal test scores for students whose parents are college graduates with a bachelor's degree. The second sample shows the SAT verbal test scores for students whose parents are high school graduates but do not have a college degree. $$\begin{array}{lccc} \text { College Grads } & & \text { High School Grads } \\ 485 & 487 & 442 & 492 \\ 534 & 533 & 580 & 478 \\ 650 & 526 & 479 & 425 \\ 554 & 410 & 486 & 485 \\ 550 & 515 & 528 & 390 \\ 572 & 578 & 524 & 535 \\ 497 & 448 & & \\ 592 & 469 & & \end{array}$$ a. Formulate the hypotheses that can be used to determine whether the sample data support the hypothesis that students show a higher population mean verbal score on the SAT if their parents attained a higher level of education. b. What is the point estimate of the difference between the means for the two populations? c. Compute the $p$ -value for the hypothesis test. d. $\quad$ At $\alpha=.05,$ what is your conclusion?

The College Board provided comparisons of Scholastic Aptitude Test (SAT) scores based on the highest level of education attained by the test taker's parents. A research hypothesis was that students whose parents had attained a higher level of education would on average score higher on the SAT. During $2003,$ the overall mean SAT verbal score was 507 (The World Almanac, 2004 ). SAT verbal scores for independent samples of students follow. The first sample shows the SAT verbal test scores for students whose parents are college graduates with a bachelor's degree. The second sample shows the SAT verbal test scores for students whose parents are high school graduates but do not have a college degree.
$$\begin{array}{lccc}
\text { College Grads } & & \text { High School Grads } \\
485 & 487 & 442 & 492 \\
534 & 533 & 580 & 478 \\
650 & 526 & 479 & 425 \\
554 & 410 & 486 & 485 \\
550 & 515 & 528 & 390 \\
572 & 578 & 524 & 535 \\
497 & 448 & & \\
592 & 469 & &
\end{array}$$
a. Formulate the hypotheses that can be used to determine whether the sample data support the hypothesis that students show a higher population mean verbal score on the SAT if their parents attained a higher level of education.
b. What is the point estimate of the difference between the means for the two populations?
c. Compute the $p$ -value for the hypothesis test.
d. $\quad$ At $\alpha=.05,$ what is your conclusion?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to support a particular claim about the population from which the sample was drawn. It involves comparing observed sample statistics to what is expected under a null hypothesis, and then determining whether any observed differences are statistically significant or could likely be due to random chance.

Null and Alternative Hypotheses

The null hypothesis typically states that there is no difference between the population parameters, while the alternative hypothesis posits that there is a difference or a specific direction of difference (such as one group having a higher mean than the other). In the context of this problem, the null hypothesis would assert that there is no difference in mean SAT scores between the two groups of students, whereas the alternative hypothesis claims that students whose parents attained a higher level of education have higher mean scores.

Independent Samples

When dealing with independent samples, the observations in one group do not affect the observations in the other group. This concept is crucial for understanding the methods used to compare means between two groups, as it determines that the standard error computation and test statistics (like the t-test statistic for two independent means) must account for the independent nature of the data.

Point Estimate

A point estimate is a single value given as an estimate of a population parameter. In the context of comparing means between two groups, the point estimate is typically the difference between the sample means. This estimate provides a direct measure of the magnitude of the observed difference between the two groups.

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis.

Significance Level and Decision Rule

The significance level, often denoted by alpha (?), is the threshold for determining whether the p-value is small enough to reject the null hypothesis. A common choice is ? = 0.05. If the computed p-value is less than or equal to this significance level, it is interpreted as sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis, leading to a conclusion that the observed effect is statistically significant.

Transcript

00:01 So in this problem, we're comparing college graduates, parents who have college degrees and the sat scores of their students compared to people that have parents who were high school graduates.

00:14 They're trying to see if there is a difference.

00:17 Well, actually, that the high school graduate students, parents that come from high school, that their score is lower than the college graduate.

00:26 So to begin with, we do want to take that data, and i put the college graduate, i'll just go cg, into list one, and the high school graduates into list two.

00:40 And we will want to find the x bar for this list, and i will just call that c for college graduate.

00:48 And i got that list to be 525.

00:51 I have the standard deviation for the college graduate parents to be 59 .42, and i have the number of students who had parents who were college graduates being 16.

01:05 And then for the other list, we have the, and i'll just call it h for high school.

01:11 The main score for those students who had parents that had graduated from high school was 487.

01:18 And the standard deviation for the high school was 51 .748.

01:27 So we had a little bit less variability here than here.

01:30 And our sample size for these was 12.

01:33 So they weren't the same.

01:36 And so we want to start out and we want to look at what the hypotheses would be for trying to show that this group of students has a significantly higher.

01:49 Mean than this one.

01:51 So we know when we assume, we would assume that the difference or that the i will go college, that the mean will say that there's no difference, that they're equal.

02:04 And alternately, that the mean for the college students is actually higher than the high school students.

02:13 Now, some people, when i teach my students, we always do this first one as equality.

02:20 There are some textbooks that will have, it could be equal, but it could also be less than, and you'll kind of do your analysis with it being equal.

02:29 But this is an alternate way to write the initial hypothesis or the null hypothesis.

02:36 And again, i usually have my students just do it as equality.

02:41 So i would assume that they're equal.

02:43 And then alternately that, do i have evidence to show that actually the mean for the college parents seems to be higher than the high school.

02:53 So we wanted to get a point estimate.

02:55 I believe that's what question b, ask you to find a point estimate for the difference.

02:59 And our point estimate is going to be the x bar of the c minus the x bar of the h.

03:05 And so when we subtract the 525 and the 487, i believe we get 38.

03:14 So that would be what our point estimate is.

03:17 So it looks like it's different from zero.

03:18 But again, we have variability.

03:21 And is this variability in our samples? is that going to cause us to think, well, you know, 38, maybe that isn't significant.

03:29 So let's do a hypothesis test.

03:32 And we have to calculate our test statistic.

03:34 And our test statistic is going to be to take the 525, and this will be a t value.

03:43 So we're doing a two sample t test.

03:47 And we'll take our 525 minus our 487, so the difference between our samples...

Please give Ace some feedback