A math teacher claims that she has developed a review course...

A math teacher claims that she has developed a review course that increases the scores of students on the math portion of the SAT exam. Based on data from the College Board, SAT scores are normally distributed with $\mu=515 .$ The teacher obtains a random sample of 1800 students, puts them through the review class, and finds that the mean SAT math score of the 1800 students is 519 with a standard deviation of $111 .$ (a) State the null and alternative hypothescs. (b) Test the hypothesis at the $\alpha=0.10$ level of significance. Is a mean SAT math score of 519 significantly higher than $515 ?$ (c) Do you think that a mean SAT math score of 519 versus 515 will affect the decision of a school admissions administrator? In other words, does the increase in the score have any practical significance? (d) Test the hypothesis at the $\alpha=0.10$ level of significance with $n=400$ students. Assume the same sample statistics. Is a sample mean of 519 significantly more than $515 ?$ What do you conclude about the impact of large samples on the $P$ -value $?$

A math teacher claims that she has developed a review course that increases the scores of students on the math portion of the SAT exam. Based on data from the College Board, SAT scores are normally distributed with $\mu=515 .$ The teacher obtains a random sample of 1800 students, puts them through the review class, and finds that the mean SAT math score of the 1800 students is 519 with a standard deviation of $111 .$
(a) State the null and alternative hypothescs.
(b) Test the hypothesis at the $\alpha=0.10$ level of significance. Is a mean SAT math score of 519 significantly higher than $515 ?$
(c) Do you think that a mean SAT math score of 519 versus 515 will affect the decision of a school admissions administrator? In other words, does the increase in the score have any practical significance?
(d) Test the hypothesis at the $\alpha=0.10$ level of significance with $n=400$ students. Assume the same sample statistics.
Is a sample mean of 519 significantly more than $515 ?$ What
do you conclude about the impact of large samples on the $P$ -value $?$

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) represents the default position that there is no effect or difference, while the alternative hypothesis (H?) represents the claim that the teacher's review course increases scores above the known mean. Formulating these hypotheses correctly establishes the framework for the testing procedure by clearly stating what is to be rejected or accepted.

Significance Level and p-Value Interpretation

The significance level, denoted by ?, is the threshold for deciding whether the observed test results are statistically significant. This threshold, such as 0.10 in this problem, represents the probability of a Type I error, which is the risk of falsely rejecting the null hypothesis when it is actually true. The p-value is then compared to ? to determine if the evidence is strong enough to reject the null hypothesis.

Test Statistic Calculation

The test statistic quantifies how far the observed sample statistic is from the null hypothesis value and is calculated using the sample mean, the hypothesized population mean, the sample standard deviation, and the sample size. This calculation often involves a z-test or t-test statistic. The computed value is then used to determine the p-value, which indicates the probability of obtaining a sample statistic as extreme or more extreme than what was observed if the null hypothesis were true.

Statistical Significance versus Practical Significance

Statistical significance refers to the mathematical likelihood that the observed result is not a product of random chance, while practical significance addresses whether the magnitude of the effect is large enough to have real-world implications. An observed difference may be statistically significant due to a large sample size but still be too small to have any meaningful impact in practice.

Impact of Sample Size on p-Value

The sample size plays a critical role in hypothesis testing. Larger samples tend to provide more precise estimates of the population parameter, which can lead to smaller p-values and increased power to detect even small differences. However, while large samples can make marginal differences statistically significant, such differences may not be practically significant, underscoring the need to consider both the p-value and the effect size.

Transcript

00:01 Hi, flip at the question here.

00:04 The match is a claim that she developed a review course that increases score of student.

00:08 It increases the score of students.

00:11 Right.

00:12 So if we take population mean, this is 550.

00:16 Population mean, this is 515 here.

00:18 So we have no hypothesis is coming up to be new equals 550.

00:22 That increases the score.

00:24 That means we have alternate deposites that is coming up to be mu is greater than 550.

00:30 Sample this sample 5 is just taken.

00:32 Sample mean.

00:33 And sample and evaluation.

00:36 So, so the test the hypothesis here at alpha is 0 .10.

00:43 You know test the hypothesis for this alpha and check that the mean set max score of 519 significantly higher than 515 so let's go for that.

00:54 We find t test statistics that's coming out to be x par negative mu over s over root n we put the values 519 root 1800 we have t test statistic and this portion that is coming out to be 1 .49 that will be 1 .49 and in technology we find the p value for this so p value is coming out to be 0 .06 3 to be in technology 0 .06 3 2 now we have alpha as given as 0 .101 and this is 3 p value is less than alpha.

01:39 Therefore, we reject null hypothesis.

01:44 So we reject null hypothesis in that case.

01:50 So that means if it will be part, is a mean that map score of pironite significantly higher than 515.

01:59 Yes, that is because we accept the alternative position.

02:01 So that is significantly higher.

02:04 Yes, significantly higher for the b part.

02:08 You can write this significantly higher.

02:16 Now c part says that you think that a means that score of 519 versus 515 will affect the decision of a school admissions administrator.

02:26 In other words, does the increase the score have any practical significance? so we can say that for the c part, yes, if the score has a practical significance.

02:36 Because 590 is significantly higher than 1515, so for three parts, it's coming out to be yes.

02:44 It has practical significance.

02:52 If you go now to d part, the d part we have, you'll test this is 4 now...

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