91Ó°ÊÓ

According to Census Bureau data, in 1998 the California population consisted of \(50.7 \%\) whites, \(6.6 \%\) blacks, \(30.6 \%\) Hispanics, \(10.8 \%\) Asians, and \(1.3 \%\) other ethnic groups. Suppose that a random sample of 1000 students graduating from California colleges and universities in 1998 resulted in the accompanying data on ethnic group. These data are consistent with summary statistics contained in the article titled "Crumbling Public School System a Threat to California's Future (Investor's Business Daily, November 12,1999\()\). $$ \begin{array}{lc} \text { Ethnic Group } & \text { Number in Sample } \\ \hline \text { White } & 679 \\ \text { Black } & 51 \\ \text { Hispanic } & 77 \\ \text { Asian } & 190 \\ \text { Other } & 3 \\ \hline \end{array} $$ Do the data provide evidence that the proportion of students graduating from colleges and universities in California for these ethnic group categories differs from the respective proportions in the population for California? Test the appropriate hypotheses using \(\alpha=.01\).

Step by step solution

01

Set Up the Hypotheses

The null hypothesis (\(H_0\)) posits that the proportions of each group in the sample are the same as their respective proportions in the population. The alternative hypothesis (\(H_a\)) is that at least one of the proportions in the sample differs from the population. So, \(H_0: P_{white} = 0.507, P_{black} = 0.066, P_{Hispanic} = 0.306, P_{Asian} = 0.108, P_{other} = 0.013\) and \(H_a: \) At least one proportion differs from the population proportion.

02

Calculate Expected Frequencies

Under the null hypothesis, we can calculate the expected frequencies for each category as the population proportion times the sample size (1000). The expected frequencies are: \(E_{white} = 0.507 * 1000 = 507\), \(E_{black} = 0.066 * 1000 = 66\), \(E_{Hispanic} = 0.306 * 1000 = 306\), \(E_{Asian} = 0.108 * 1000 = 108\), and \(E_{other} = 0.013 * 1000 = 13\).

03

Compute Chi-Square Statistic

The test statistic for a test of proportion is the chi-square statistic, calculated as \(\chi^2 = \sum (O-E)^2 / E\), where sum is over all categories, \(O\) denotes observed frequency, and \(E\) denotes expected frequency. The chi-square statistic is calculated separately for each category and then summed.

04

Determine Critical Value

The critical value for the chi-square distribution with 4 degrees of freedom (5 groups - 1) at an \(\alpha = 0.01\) level of significance is approximately 13.28.

05

Make a Decision

If the calculated chi-square value is greater than the critical value, reject the null hypothesis, indicating that the distribution of ethnic group among college graduates significantly differs from the distribution in the general population. If the calculated chi-square value is less than or equal to the critical value, fail to reject the null hypothesis, which suggests that the distribution among college graduates does not significantly differ from that in the population.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ethnic Group Distribution

Ethnic group distribution refers to how different ethnicities are represented within a specific population or sample. In this exercise, we focus on the distribution in California's population in 1998. This distribution is essential for understanding both demographic composition and changes over time. For California, the distribution was as follows:

• 50.7% Whites
• 6.6% Blacks
• 30.6% Hispanics
• 10.8% Asians
• 1.3% Other ethnic groups

A critical factor to consider is how the sample of graduating students compared with this distribution. Understanding this allows us to assess if the representation in the graduating population differs significantly from the general population's makeup at the time. To effectively evaluate this, we perform statistical testing to see if any disparities exist between the populations.

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics used to determine if there is enough evidence to reject a preconceived notion about a population. Here, it's applied to test whether the sample distribution of ethnicities among California college graduates in 1998 differs from the state's overall population at the time. In hypothesis testing, you begin with the null hypothesis ( $H_0$ ), suggesting no difference between the sample and the population proportions. The alternative hypothesis ( $H_a$ ) implies at least one group in the sample differs from the population. We use observable data (our sample) to make inferences about the unobservable population parameter. The chi-square test assesses these differences by examining the observed frequencies against expected frequencies calculated from the population proportions. A systematic approach involves several steps:

• Establish the null and alternative hypotheses.
• Calculate expected frequencies under $H_0$ .
• Compute the chi-square statistic (χ^2 =∑ (O−E)^2/E) to measure differences between observed ( $O$ ) and expected ( $E$ ) values.
• Evaluate this statistic against a critical value determined by the level of significance and degrees of freedom (number of groups minus one).
• Decide whether to accept or reject $H_0$ based on the comparison.

The goal is to provide evidence towards either maintaining the status quo or supporting an alternative view based on the sample data.

Significance Level

The significance level in hypothesis testing is a pivotal part of determining whether the results of your test are statistically significant. In plain terms, it is the probability of rejecting the null hypothesis when it is actually true. It is denoted by alpha (\(\alpha\)), and in this case, it is set at 0.01. Using a significance level of 0.01 implies that you are willing to accept a 1% chance of being wrong when you reject the null hypothesis. This stringent level underscores the need for a high degree of confidence in our test results. A lower significance level means you require more substantial evidence to reject the null hypothesis. Choosing the right significance level depends on the context, including the potential consequences of making an error:

• A low \(\alpha\) (e.g., 0.01) is used in cases where consequences of a Type I error (false positive) are significant.
• A higher \(\alpha\) (e.g., 0.05) might be acceptable in preliminary studies where exploration takes precedence over accuracy.

In any chi-square test, comparing the computed chi-square statistic against the critical value (corresponding to the chosen \(\alpha\) and degrees of freedom) helps make an informed decision on the hypothesis.