The psychology department is gradually changing its...

The psychology department is gradually changing its curriculum by increasing the number of online course offerings. To evaluate the effectiveness of this change, a random sample of $n=36$ students who registered for Introductory Psychology is placed in the online version of the course. At the end of the semester, all students take the same final exam. The average score for the sample is $M=76 .$ For the general population of students taking the traditional lecture class, the final exam scores form a normal distribution with a mean of $\mu=71$. a. If the final exam scores for the population have a standard deviation of $\sigma=12,$ does the sample provide enough evidence to conclude that the new online course is significantly different from the traditional class? Use a two-tailed test with $\alpha=.05$ b. If the population standard deviation is $\sigma=18,$ is the sample sufficient to demonstrate a significant difference? Again, use a two-tailed test with $\alpha=.05$ c. Comparing your answers for parts a and b, explain how the magnitude of the standard deviation influences the outcome of a hypothesis test.

The psychology department is gradually changing its curriculum by increasing the number of online course offerings. To evaluate the effectiveness of this change, a random sample of $n=36$ students who registered for Introductory Psychology is placed in the online version of the course. At the end of the semester, all students take the same final exam. The average score for the sample is $M=76 .$ For the general population of students taking the traditional lecture class, the final exam scores form a normal distribution with a mean of $\mu=71$.
a. If the final exam scores for the population have a standard deviation of $\sigma=12,$ does the sample provide enough evidence to conclude that the new online course is significantly different from the traditional class? Use a two-tailed test with $\alpha=.05$
b. If the population standard deviation is $\sigma=18,$ is the sample sufficient to demonstrate a significant difference? Again, use a two-tailed test with $\alpha=.05$
c. Comparing your answers for parts a and b, explain how the magnitude of the standard deviation influences the outcome of a hypothesis test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves formulating a null hypothesis (which typically represents the status quo or no effect) and an alternative hypothesis (representing the change or effect of interest) and then using a test statistic to determine whether there is enough evidence to reject the null hypothesis at a predetermined significance level.

Null and Alternative Hypotheses

The null hypothesis (H0) posits that there is no significant difference between groups or conditions—in this case, that the online course’s exam scores are the same as those for the traditional class. The alternative hypothesis (Ha) suggests that there is a significant difference. Setting up these hypotheses is crucial because they guide the direction and interpretation of the statistical test.

Z-Test

A Z-test is used when the population standard deviation is known and the sampling distribution of the sample mean is approximately normal. The Z-test compares the observed sample mean to the population mean by converting the difference into a standardized score, allowing for the calculation of a p-value to assess the significance of the result.

Standard Error

The standard error quantifies the amount of variability in the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size (?/?n). A smaller standard error indicates that the sample mean is likely to be close to the true population mean, making the results of the test more precise.

Significance Level and p-Value

The significance level (?), such as 0.05, represents the threshold for making a Type I error, which is incorrectly rejecting the null hypothesis when it is true. The p-value, calculated from the test statistic, indicates the probability of observing a sample mean as extreme as the one obtained if the null hypothesis were true. If the p-value is less than the significance level, the null hypothesis is rejected.

Impact of Population Standard Deviation

The magnitude of the population standard deviation directly affects the standard error of the sample mean. A larger standard deviation increases the standard error, which makes the test statistic smaller for the same difference between the sample mean and population mean, potentially leading to a non-significant result. Conversely, a smaller standard deviation decreases the standard error, increasing the magnitude of the test statistic and making it easier to detect significant differences.

Transcript

00:01 So we have a sample size of 36, and they get an x bar for the final exam of 76.

00:09 And we're to assume that the population mean for final exams is 71.

00:15 And we want to know if this score is basically significantly different.

00:19 So we would be assuming that the class really has a mean of 71, and alternately that the mean is different from 71.

00:30 So if i draw a little picture of our distribution, our sampling distribution is for a sample size of 36.

00:37 And that 36 is going to be relatively large enough for this distribution to be approximately normal.

00:45 And we're going to center it at 71.

00:48 However, they're getting a 76.

00:50 Now, we haven't taken into account standard deviation yet.

00:54 And so we're five points higher and we go five points lower.

00:59 Whoops.

00:59 She's 76 there.

01:01 That area combined together is our p value.

01:05 So in part a, we're to assume that the standard deviation of the population of individuals is 12.

01:11 So we want to find what's the likelihood of getting an x bar from this distribution.

01:18 The sample size is 36 and having the mean come out to be greater than or equal to 76.

01:24 Now, we're doing a two -tail test, so we're saying it's just as likely if this disdivision.

01:29 Distribution is centered at 71 for us to get a mean of 66 or lower.

01:35 So now we convert this to a test statistic.

01:39 It will be a z value since our population standard deviation is known.

01:44 And we take our 76 minus 71.

01:48 And then we divide it by the population for individuals divided by the square root of 36, which is the sample size.

01:58 So let's see what we get here.

02:00 We end up getting the z value here is we have 5, 5 divided by, and then in parentheses 12 divided by 6 or 2.

02:15 And so we get 2 .5 for the z value.

02:19 Now we can use our book and find what this area is and then double it to find the lower area.

02:26 Or also, i'm going to end up using my normal cdf on my calculator, since i don't actually have the table sitting right here...

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