Question

Do teachers find their work rewarding and satisfying? The article "Work- Related Attitudes" (Psychological Reports \([1991]: 443-450)\) reported the results of a survey of random samples of 395 elementary school teachers and 266 high school teachers. Of the elementary school teachers, 224 said they were very satisfied with their jobs, whereas 126 of the high school teachers were very satisfied with their work. Based on these data, is it reasonable to conclude that the proportion very satisfied is different for elementary school teachers than it is for high school teachers? Test the appropriate hypotheses using a \(.05\) significance level.

Short answer

At a \(.05\) significance level, the claim that the proportion of job satisfaction differs between elementary school teachers and high school teachers is not supported by the provided data.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis, \(H_0\), is that the proportions of elementary school teachers and high school teachers who are very satisfied are the same. The alternative hypothesis, \(H_a\), is that the proportions are different. Formally, these hypotheses can be stated as: \(H_0: p_1 = p_2) \ and \( Ha: p_1 ≠ p_2 \).

Calculate the Sample Proportions and Their Difference

The sample proportion of elementary school teachers who are very satisfied is \(p_1 = 224/395 = 0.567\). The sample proportion of high school teachers who are very satisfied is \(p_2 = 126/266 = 0.474\). The difference between the two sample proportions is \(\hat{p}_1 - \hat{p}_2 = 0.567 - 0.474 = 0.093\). The pooled proportion, under the null hypothesis that \(p_1 = p_2\), is \(\hat{p} = (x_1 + x_2) / (n_1 + n_2) = (224 + 126) / (395 + 266) = 0.527\).

Calculate the Standard Error

The standard error of the difference between the two proportions, under the null hypothesis, is \(\sqrt{\hat{p}(1-\hat{p})(1/n_1 + 1/n_2)} = \sqrt{0.527*(1-0.527)*(1/395 + 1/266)} = 0.0481\).

Calculate the Test Statistic and the P-Value

The test statistic is \((\hat{p}_1 - \hat{p}_2) / SE = 0.093 / 0.0481 = 1.932\). This statistic follows approximately a standard normal distribution under the null hypothesis. The two-tailed P-value can be found by looking up this statistic in a standard normal table, which gives a P-value of 0.053.

Draw a Conclusion

At a \(.05\) significance level, since the P-value \(0.053 > 0.05\), we do not reject the null hypothesis. It is concluded that at this level of significance, there is not enough evidence to support the claim that the proportion of job satisfaction is different between the elementary school and high school teachers.

Key concepts

Null Hypothesis and Alternative Hypothesis

When we speak of null hypothesis and alternative hypothesis, we are discussing one of the fundamental components in statistical hypothesis testing. The null hypothesis, denoted as H₀, represents a statement of no effect or no difference; it is the assertion that any observed differences in a set of data are due simply to random chance. In our exercise related to teacher job satisfaction, the null hypothesis would state that the level of job satisfaction among elementary school teachers is identical to that of high school teachers.

The alternative hypothesis, denoted as H_a or H₁, is what you would conclude in the case where there is significant statistical evidence to suggest that the null hypothesis is not true. In this context, the alternative hypothesis claims that there is a difference in job satisfaction between the two groups of teachers. These hypotheses lay the groundwork for conducting the test and interpreting the results.

Sample Proportion Calculation

The concept of sample proportion is a measurement used to express the fraction of the sample with a certain characteristic. With regard to teacher satisfaction, this proportion represents the number of teachers who reported being very satisfied divided by the total number of teachers surveyed. Representing the sample proportion for elementary teachers as p₁ and for high school teachers as p₂, we calculate these by dividing the number of satisfied respondents by the total number surveyed in each category. It is the difference between these two sample proportions that we analyze to determine if there's statistical evidence to support the alternative hypothesis.

Standard Error of Difference

The standard error of the difference between two proportions gives us a sense of the variability we'd expect to find between these two sample proportions if the null hypothesis were true—essentially, serving as a way to normalize our comparison. By calculating this standard error, we obtain an estimate of the standard distance that the observed sample difference is away from zero difference (if the null hypothesis is true). To calculate the standard error, we use the pooled proportion (assuming the null hypothesis is correct) and the sizes of both samples. This value is crucial as it relates to the calculation of the test statistic, which we'll use to infer whether the observed difference between our sample proportions is statistically significant or not.

P-value Interpretation

The P-value is a probabilistic measurement that tells us the likelihood of obtaining an observed outcome, or one more extreme, given that the null hypothesis is true. In more colloquial terms, it's the chance of seeing the data you have if there were really no underlying effect. P-value interpretation is key in hypothesis testing: a small P-value (<0.05 as a common threshold) implies that the observed data would be unlikely under the null hypothesis, leading us to reject H₀ in favor of H_a. Conversely, a large P-value suggests that the observed data are consistent with the null hypothesis, so we do not reject H₀. In our exercise, the P-value exceeds the 0.05 threshold, so we conclude that there's not enough evidence to say that satisfaction levels for elementary and high school teachers are different.

Job Satisfaction in Teachers

The concept of job satisfaction can be multifaceted, especially within a teaching context. It encapsulates elements from emotional satisfaction to professional fulfillment. Comparing job satisfaction between different educational levels, such as elementary and high school teachers, can be influenced by various factors including work environment, resources, and student interaction. The exercise provided offers insight into statistical evidence, or the lack thereof, regarding varying levels of job satisfaction between different teaching positions. Understanding the factors that contribute to job satisfaction is crucial, as it can impact teacher retention, student success, and the overall health of educational institutions.