Question

The report "California's Education Skills Gap: Modest Improvements Could Yield Big Gains" (Public Policy Institute of California, April \(16,2008,\) www.ppic.org) states that nationwide, \(61 \%\) of high school graduates go on to attend a two-year or four-year college the year after graduation. The proportion of high school graduates in California who go on to college was estimated to be \(0.55 .\) Suppose that this estimate was based on a random sample of 1,500 California high school graduates. Is it reasonable to conclude that the proportion of California high school graduates who attend college the year after graduation is different from the national figure? (Hint: Use what you know about the sampling distribution of \(\hat{p}\). You might also refer to Example \(8.5 .)\)

Short answer

To make a conclusion, calculate the Z score using the given data. If the calculated Z-property lies between -1.96 and 1.96, do not reject the null hypothesis, meaning the proportion of California high school students going to college is not significantly different from the national average. If the Z score lies outside this range, reject the null hypothesis, pointing to a significant difference.

Step-by-step solution

State the Null Hypothesis and Alternative Hypothesis

The Null Hypothesis (\(H_0\)) states that the proportion of California high school graduates going to college the year after graduation equals the expected (national) proportion. So, \(H_0: p = 0.61\)\nThe Alternative Hypothesis (\(H_1\)) states that the proportion of California high school graduates going to college the year after graduation is different from the expected (national) proportion. In this case, \(H_1: p \neq 0.61\)

Calculate The Test Statistic

The formula for calculating the Z-score in hypothesis tests for proportions is \[Z =\frac{\(p_{\text{sample}} - p_{\text{null}}\)}{\sqrt{\(p_{\text{null}} * (1 - p_{\text{null}})\) / n}} \]\nSubstitute \(p_{\text{sample}} = 0.55\), \(p_{\text{null}} = 0.61\), and \(n = 1500\) into the equation. This will give the Z score value which represents how far and in what direction the sample mean deviates from the population mean.

Interpret the Test Statistic

Once the Z-score is calculated, compare this with the standard normal distribution. A Z-score of less than -1.96 or more than 1.96 will be considered as statistically significant at \(\alpha = 0.05\) level. If the Z-score is outside this range, then there is a statistically significant difference between the two groups, and the null hypothesis would be rejected.

Key concepts

Null Hypothesis

When diving into the realm of hypothesis testing in statistics, the concept of the null hypothesis is foundational. In essence, the null hypothesis, denoted as \(H_0\), represents a default position: that there is no effect or no difference. It's essentially our starting point for analysis.

For example, if we're looking at the proportion of high school graduates who go on to attend college, our null hypothesis might state that there is no difference between the proportion at the California level compared to the national figure – \(H_0: p = 0.61\). It is crucial to outline the null hypothesis clearly because it sets the stage for the ensuing statistical test and is the precise claim that evidence from the sample data will either lend support to or cast doubt upon.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis \(H_1\) or \(H_a\) is a researcher's assertion about the population parameter that is competing with the null hypothesis. The alternative hypothesis posits that there is a statistically significant effect or difference. This could mean an assertion of a difference in proportions, mean values or a relationship between variables.

For the illustration regarding school graduates, our alternative hypothesis would suggest that Californian school graduates' college attendance rate differs from the national proportion – \(H_1: p eq 0.61\). It’s expressed in such a way that it covers all possible outcomes that are not described by the null hypothesis.

Test Statistic

The test statistic is a crucial component that tells us how far our sample statistic is from the null hypothesis value, when measured in standard errors. The calculation of the test statistic incorporates both the sample data and the assumed population parameter under the null hypothesis.

To compute the test statistic, you would use the formula indicated in the solution, where the numerator encapsulates the difference between the sample statistic and null hypothesis value and the denominator adjusts for sample size. In this case, it’s a Z-score, which we’ll delve deeper into in the following section.

Z-score

The Z-score is one iteration of a test statistic, which is used when comparing sample data to a known population standard. It reveals how many standard deviations an element is from the mean. When you calculate the Z-score for a hypothesis test, you’re essentially transforming your data point (in this scenario, the proportion of graduates attending college) into a score that reflects the number of standard deviations away from the average value posited by the null hypothesis.

A Z-score can tell us where a proportion sits within the normal distribution and is instrumental in determining whether the observed data are within the realm of normal variability or statistically unusual. For hypothesis testing, specific Z-score thresholds determine whether we reject or fail to reject the null hypothesis.

Sampling Distribution

The concept of a sampling distribution is perhaps less intuitive than some other statistical concepts, but it's incredibly powerful. A sampling distribution isn't based on just one sample; instead, it's the distribution of a statistic (like a mean or proportion) that we'd get if we took many samples and calculated the statistic for each one.

In the context of hypothesis testing, we’re often using the sampling distribution of the test statistic (such as the Z-score). It describes how our test statistic would behave if we repeated our sampling many times. This distribution is essential for understanding the variability inherent in sampling and for calculating probabilities that direct our decision-making concerning the null hypothesis.