Question

Typically, only very brave students are willing to speak out in a college classroom. Student participation may be especially difficult if the individual is from a different culture or country. The article "An Assessment of Class Participation by International Graduate Students" \((\) Journal of College Student Development \([1995]: 132-\) 140) considered a numerical "speaking-up" scale, with possible values from 3 to 15 (a low value means that a student rarely speaks). For a random sample of 64 males from Asian countries where English is not the official language, the sample mean and sample standard deviation were \(8.75\) and \(2.57\), respectively. Suppose that the mean for the population of all males having English as their native language is \(10.0\) (suggested by data in the article). Does it appear that the population mean for males from non-English-speaking Asian countries is smaller than \(10.0 ?\)

Short answer

The short answer should be whether or not we reject the null hypothesis that the population mean for males from non-English-speaking Asian countries is 10.0, based on the computed t-statistic and the critical value.

Step-by-step solution

State the Hypotheses

The null hypothesis \(H_0\) is that the population mean \(\mu\) is equal to 10, i.e., \(\mu = 10\). The alternative hypothesis \(H_1\) is that the population mean \(\mu\) is less than 10, i.e., \(\mu < 10\).

Calculate the T-Statistic

The t-statistic is given by the formula \(t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}\) where \(\bar{X}\) is the sample mean, \(\mu_0\) is the population mean under the null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the sample size. Substituting the given values, we get \(t = \frac{8.75 - 10}{2.57 / \sqrt{64}}\).

Determine the Degrees of Freedom

The degrees of freedom for a single sample t-test is \(df = n - 1\). So here, \(df = 64 - 1 = 63\).

Find the Critical Value

We need to find the critical t-value for a one-tailed test (since our alternative hypothesis is \(\mu < 10\)) at a significance level of 0.05 (typically used in social sciences) with 63 degrees of freedom. You can use a t-distribution table or an online calculator to find this value.

Make a Decision

If the calculated t-statistic is less than the critical value, we reject the null hypothesis. If it's greater than or equal to the critical value, we fail to reject the null hypothesis. Based on this rule, decide whether it appears that the population mean for males from non-English-speaking Asian countries is smaller than 10.0.

Key concepts

Null Hypothesis

Understanding the null hypothesis is crucial when you're delving into hypothesis testing. It's the default claim concerning the population parameter, which in our exercise is the population mean speaking-up score for males with English as their native language. Here, the null hypothesis (\(H_0\)) posits that there is no difference between the mean score of our sample of Asian males and the established mean of 10.0. It essentially states, 'until proven otherwise, we assume that the mean speaking-up score for Asian males from non-English speaking countries is the same as English-speaking nationals.' The significance of the null hypothesis lies in its role as a starting point for statistical testing, which challenges whether there's enough evidence to support an alternative claim.

T-statistic

The t-statistic is a type of standard score that lets you compare a sample mean to the population mean in a hypothesis test, especially when the population variance is unknown. It's calculated using the formula \(t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}\), where \(\bar{X}\) represents the sample mean, \(\mu_0\) the population mean under the null hypothesis, \(s\) the sample standard deviation, and \(n\) the sample size. For our exercise, the t-statistic measures how many standard error units the sample mean is below the population mean of 10.0. If the t-statistic falls within a certain range, it can indicate that the difference observed might be due to chance.

Degrees of Freedom

In statistics, the concept of degrees of freedom refers to the number of values that are free to vary in the calculation of a statistic without breaking any constraints. It's essential for determining the appropriateness of a statistical model and choosing the right critical value from the t-distribution. For most t-tests, including our exercise, the degrees of freedom are calculated as the sample size minus one (\(df = n - 1\)). This value comes into play when we’re using the t-distribution to determine the critical value, against which we compare the t-statistic.

Critical Value

The critical value is a point on the scale of the test statistic—beyond which we would reject the null hypothesis. In the context of our t-test, it’s the t-value that corresponds to a certain level of significance (often 0.05 for a 95% confidence level) according to the degrees of freedom. It forms the threshold for deciding whether the t-statistic indicates a statistically significant difference. For a one-tailed test, we'd only look for a critical value on one end of the t-distribution, reflecting our directional hypothesis that the Asian male students speak up less frequently than male students for whom English is the native language.

Sample Mean

The sample mean (\(\bar{X}\)) is simply the average score from our sample group. In hypothesis testing, it serves as the estimate of our population parameter—in this case, the mean speaking-up score for Asian males. By comparing the sample mean to the population mean under null hypothesis orthodoxy, we can begin to draw conclusions about the greater population. In our example, a sample mean of 8.75 suggests that Asian males from non-English-speaking countries may have a lower average speaking-up score compared to the population mean of 10.0 for English-speaking males.

Sample Standard Deviation

When it comes to the sample standard deviation (\(s\)), think of it as a measure of how much variability or dispersion there is from the sample mean. It's critical in calculating the t-statistic because it affects the standard error of the mean. A larger standard deviation indicates more variability in the sample scores and may affect the reliability of the sample mean as a representation of the population mean. In our example, a standard deviation of 2.57 reflects the extent to which individual speaking-up scores deviate from the sample mean.

One-Tailed Test

The one-tailed test is a form of hypothesis test where the region of rejection for the null hypothesis is on only one side of the sampling distribution. It’s used when we have a specific direction in our alternative hypothesis. For instance, our question is concerned with whether the speaking-up score is 'less than' a certain value—not just different. This directionality means we only look at the lower tail of the t-distribution to find our critical value and decide if there is enough evidence to support the claim that Asian males from non-English speaking countries speak up less frequently than their English-speaking counterparts in a college classroom setting.