91Ó°ÊÓ

The state of California is working very hard to ensure that all elementary age students whose native language is not English become proficient in English by the sixth grade. Their progress is monitored each year using the California English Language Development test. The results for two school districts in southern California for the 2003 school year are given in the accompanying table. \(^{\star}\) Do the data indicate a significant difference in the 2003 proportions of students who are fluent in English for the two districts? Use \(\alpha=.01\) $$\begin{array}{lcc} \hline \text { District } & \text { Riverside } & \text { Palm Springs } \\ \hline \text { Number of students tested } & 6124 & 5512 \\ \text { Percentage fluent } & 40 & 37 \\ \hline \end{array}$$

Step by step solution

01

Set Up Hypotheses

We need to test the null hypothesis that there is no significant difference in proportions of students fluent in English between the two districts, Riverside and Palm Springs. Let \( p_1 \) be the proportion of students fluent in English in Riverside, and \( p_2 \) for Palm Springs. The null hypothesis \( H_0 \) is \( p_1 = p_2 \). The alternative hypothesis \( H_a \) is \( p_1 eq p_2 \).

02

Calculate Sample Proportions

For Riverside, the number of fluent students is \( 0.40 \times 6124 = 2449.6 \approx 2450 \). So, the sample proportion \( \hat{p}_1 = \frac{2450}{6124} = 0.40 \). For Palm Springs, \( 0.37 \times 5512 = 2039.44 \approx 2039 \). Hence, \( \hat{p}_2 = \frac{2039}{5512} = 0.37 \).

03

Calculate the Pooled Proportion

To compute the standard error, first find the pooled proportion \( \hat{p} \) as follows: \[ \hat{p} = \frac{2450 + 2039}{6124 + 5512} = \frac{4489}{11636} \approx 0.3857. \]

04

Compute the Standard Error

Using the pooled proportion \( \hat{p} \), calculate the standard error (SE) for the difference in proportions: \[ SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.3857 \times 0.6143 \left( \frac{1}{6124} + \frac{1}{5512} \right)} \approx 0.0093. \]

05

Calculate the Test Statistic

The test statistic for the difference in proportions is: \[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.40 - 0.37}{0.0093} \approx 3.23. \]

06

Determine Critical Value or P-value

At a significance level \( \alpha = 0.01 \), for a two-tailed test, the critical value of \( Z \) is approximately \( \pm 2.576 \). Alternatively, use the standard normal distribution to find the p-value for \( Z = 3.23 \).

07

Make a Decision

Since the calculated \( Z \) value of 3.23 exceeds the critical value of 2.576, or equivalently, the p-value is less than 0.01, reject the null hypothesis.

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Key Concepts

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

Proportions

In hypothesis testing, "proportions" refer to the fraction or percentage of a certain group within a sample that exhibits a particular characteristic. For instance, in this exercise, we are interested in the proportion of students who are proficient in English in two different school districts.
To calculate this, you take the number of students fluent in English and divide it by the total number of students tested.

• For Riverside, the proportion is calculated as \( \hat{p}_1 = \frac{2450}{6124} = 0.40 \).
• For Palm Springs, it is \( \hat{p}_2 = \frac{2039}{5512} = 0.37 \).

These proportions are essential as they serve as the sample estimates of the population proportions. Evaluating these proportions allows us to compare the two districts and identify potential differences in English proficiency rates.

Significance Level

The significance level, often denoted by the Greek letter \( \alpha \), is a critical concept in hypothesis testing. It represents the threshold for determining whether an observed effect is statistically significant.
The significance level is the probability of rejecting the null hypothesis when it is actually true. In other words, it is the risk we are willing to take of making a Type I error.
For this exercise, the significance level is set at \( \alpha = 0.01 \).

• This means we only accept a result as significant if there is less than a 1% probability that the observed difference happened by chance.
• Such a low significance level is often used when high certainty is required, minimizing the chance of incorrectly claiming a difference between the districts.

Choosing an appropriate significance level is crucial in ensuring the reliability of your conclusions.

Pooled Proportion

The pooled proportion is an averaged proportion of two or more samples. It is used when the hypothesis test aims to compare the proportions from different groups. In this exercise, it assists in calculating the standard error.
The pooled proportion \( \hat{p} \) combines data from both Riverside and Palm Springs districts:

• \[ \hat{p} = \frac{2450 + 2039}{6124 + 5512} = \frac{4489}{11636} \approx 0.3857. \]

Using the pooled proportion instead of separate proportions helps in standardizing differences for a more accurate calculation of the test statistic. By averaging the data, the pooled proportion acts as a common reference point, improving the precision of the comparison between the two districts.

Standard Error

Standard Error (SE) measures how much we expect the sample proportion to vary from the true population proportion. It helps in determining the reliability and precision of the sample statistics.
In the context of proportions, the formula for standard error when using a pooled proportion is:

• \[ SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.3857 \times 0.6143 \left( \frac{1}{6124} + \frac{1}{5512} \right)} \approx 0.0093. \]

This calculated standard error shows the variability one might expect when estimating the difference in proportions from random samples of these sizes. A smaller standard error indicates that our sample proportions are likely to be close to the true population proportions, enhancing the confidence in the hypothesis test's results.