Self-Esteem Scores In a study of a group of women science...

Self-Esteem Scores In a study of a group of women science majors who remained in their profession and a group who left their profession within a few months of graduation, the researchers collected the data shown here on a self-esteem questionnaire. At $\alpha=0.05,$ can it be concluded that there is a difference in the self-esteem scores of the two groups? Use the $P$ -value method. $$ \begin{array}{ll}{\text { Leavers }} & {\text { Stayers }} \\ {\bar{X}_{1}=3.05} & {\bar{X}_{2}=2.96} \\ {\sigma_{1}=0.75} & {\sigma_{2}=0.75} \\ {n_{1}=103} & {n_{2}=225}\end{array} $$

Self-Esteem Scores In a study of a group of women science majors who remained in their profession and a group who left their profession within a few months of graduation, the researchers collected the data shown here on a self-esteem questionnaire. At $\alpha=0.05,$ can it be concluded that there is a difference in the self-esteem scores of the two groups? Use the $P$ -value method.
$$
\begin{array}{ll}{\text { Leavers }} & {\text { Stayers }} \\ {\bar{X}_{1}=3.05} & {\bar{X}_{2}=2.96} \\ {\sigma_{1}=0.75} & {\sigma_{2}=0.75} \\ {n_{1}=103} & {n_{2}=225}\end{array}
$$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical process used to make inferences or draw conclusions about a population based on sample data. It involves formulating a null hypothesis (a statement of no effect or difference) and an alternative hypothesis (representing the presence of an effect or difference), then using sample data to decide whether to reject the null hypothesis in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis is a statement that there is no effect or difference between groups, while the alternative hypothesis posits that there is a significant effect or difference. These hypotheses set the foundation for the testing process by providing a clear benchmark against which sample data can be compared.

P-Value

The P-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. It measures the strength of the evidence against the null hypothesis; a small P-value indicates that such an extreme observed outcome would be unlikely if the null hypothesis were true.

Significance Level

The significance level, often denoted by alpha (?), is the threshold set by the researcher for determining whether the P-value indicates strong enough evidence to reject the null hypothesis. Commonly used significance levels are 0.05 or 0.01, and if the P-value is less than or equal to the significance level, the null hypothesis is rejected.

Test Statistic

The test statistic is a standardized value calculated from sample data, used in determining how far the sample statistic is from the population parameter under the null hypothesis. Its value is compared against a critical value or used to compute the P-value, and it helps in making a decision about whether to reject the null hypothesis.

Two-Sample Comparison of Means

In comparing the means of two independent groups, the two-sample test is used to determine if there is a statistically significant difference between the group means. This methodology involves calculating a difference between the sample means, finding its standard error, and then using this information to conduct a hypothesis test.

Z-Test

A Z-test is a type of hypothesis test that is used when the sampling distribution of the test statistic is normally distributed and the population standard deviations are known. It involves converting the difference between the sample statistic and the hypothesized parameter into a standardized Z-score which follows a standard normal distribution.

Transcript

00:01 Okay, so we want to compute and compare the self -esteem of women who decided to leave their career or stay in their career, specifically women and graduated in science.

00:12 We need to use a p -value method.

00:14 So if you can refresh on the p -value method, you can go at page 431 of your book.

00:20 Let's get started.

00:21 So first step, just like most math problem, we need to gather our data.

00:26 So women who left their profession, the mean sample score for their.

00:32 So the mean self -esteem score for the sample is 3 .05, 3 .05.

00:40 Standard deviation of the population, so then the sample, the population is 0 .75.

00:46 And the number of women in that sample is 103, which is greater than 30.

00:55 We're going to find the same information for women who stayed in their profession.

01:03 So the mean self -esteem score for the sample is 2 .96.

01:11 They both have the same standard deviation.

01:13 Both population have same standard deviation.

01:16 And the sample size of women who stayed in their profession is 225, which is also greater than 30.

01:23 Perfect.

01:26 Now we need to write down our hypothesis.

01:31 So we have our now hypothesis and our alternate hypothesis.

01:34 We're going to have for the standard normal hypotheses, that mean we think the average, self -s -discount for both population is the same, and alternative hypotheses, we say that the means self -esteem score for both population is different.

01:52 So we have everything we need.

01:54 Now, we're going to step two.

01:57 We're going to find z.

01:58 Z is the same formula of page 491.

02:02 So we have x1 bar minus x2 bar.

02:06 Take the difference in population mean we divide by the variance square the variance divided by this number by the sample size which we had the other standard deviation divided by the other sample size and we put everything in this nice square root remember this is h0 therefore we know that the difference in population mean is zero we can write down all those nice numbers.

02:45 So we have 3 .05 minus 2 .96 over 0 .75 squared over 103 plus 0 .75 square over 225.

03:03 You put everything in a square room...

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