A Local School Board Wants To Determine The Proportion Of Households In The District That Would Support A Proposal To Start The School Year A Week Earlier. They Ask A Random Sample Of 100 Households Whether They Would Support The Proposal, And 62

A Local School Board's Proposal: Estimating the Proportion of Households in Support

In the world of education, decisions made by local school boards can have a significant impact on the community. One such decision is the proposal to start the school year a week earlier. To gauge the public's opinion on this proposal, a local school board in a district decided to conduct a random sample survey of households. The goal of this survey was to determine the proportion of households in the district that would support the proposal. In this article, we will delve into the world of statistics and explore how the school board can estimate the proportion of households in support of the proposal.

The school board conducted a random sample survey of 100 households in the district. Out of these 100 households, 62 responded in support of the proposal to start the school year a week earlier. This means that the proportion of households in support of the proposal is 62/100 or 0.62. However, this is just a sample proportion and may not accurately represent the true proportion of households in the district.

To estimate the population proportion, we need to consider the concept of sampling error. The sampling error is the difference between the sample proportion and the population proportion. In this case, the sampling error is 0.62 - p, where p is the true population proportion. We can use the formula for the standard error of the proportion to estimate the sampling error:

SE = sqrt(p(1-p)/n)

where SE is the standard error, p is the true population proportion, and n is the sample size.

Using the sample proportion (0.62) as an estimate of the population proportion, we can calculate the standard error:

SE = sqrt(0.62(1-0.62)/100) = sqrt(0.62(0.38)/100) = sqrt(0.2356/100) = sqrt(0.002356) = 0.0485

To construct a confidence interval for the population proportion, we need to use the standard error and the critical value from the standard normal distribution. Let's assume we want to construct a 95% confidence interval. The critical value for a 95% confidence interval is approximately 1.96.

Lower Bound

The lower bound of the confidence interval is:

p - 1.96(SE) = 0.62 - 1.96(0.0485) = 0.62 - 0.0953 = 0.5247

Upper Bound

The upper bound of the confidence interval is:

p + 1.96(SE) = 0.62 + 1.96(0.0485) = 0.62 + 0.0953 = 0.7153

The 95% confidence interval for the population proportion is (0.5247, 0.7153). This means that we are 95% confident that the true population proportion of households in support of the proposal to start the school year a week earlier is between 0.5247 and 0.7153. While this interval provides a range of possible values, it is essential to note that the true population proportion may not lie within this interval.

The confidence interval suggests that the true population proportion of households in support of the proposal is likely to be between 52.47% and 71.53%. This means that the school board can expect that between 52.47% and 71.53% of households in the district would support the proposal to start the school year a week earlier. However, it is crucial to note that this is just an estimate and may not accurately represent the true population proportion.

There are several limitations to this study. Firstly, the sample size is relatively small (n = 100), which may lead to a larger sampling error. Secondly, the survey may not have been representative of the entire district, as it only included a random sample of households. Finally, the survey may have been subject to biases, such as non-response bias or social desirability bias.

To improve the accuracy of the estimate, the school board could consider increasing the sample size or using a more representative sample. Additionally, they could use more advanced statistical methods, such as regression analysis or propensity scoring, to control for potential confounding variables. By taking these steps, the school board can increase the accuracy of their estimate and make more informed decisions about the proposal to start the school year a week earlier.

• [1] National Center for Education Statistics. (2020). Digest of Education Statistics 2020.
• [2] American Community Survey. (2020). American Community Survey 2020.
• [3] Pew Research Center. (2020). Public Opinion on Education Policy.

The data used in this study is available upon request.