Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Births A random sample of 860 births in New York State included 426 boys. Construct a 95% confidence interval estimate of the proportion of boys in all births. It is believed that among all births, the proportion of boys is 0.512. Do these sample results provide strong evidence against that belief?

Verified step by step guidance

1. Step 1: Identify the given values. From the problem, the sample size (n) is 860, the number of boys (x) is 426, and the confidence level is 95%. The hypothesized population proportion (p₀) is 0.512.
2. Step 2: Calculate the sample proportion (p̂). The sample proportion is given by the formula: $\mathrm{p̂}=\frac{x}{n}$. Substitute the values of x and n into the formula.
3. Step 3: Determine the critical value (z*) for a 95% confidence level. Use a standard normal distribution table or calculator to find the z* value corresponding to a 95% confidence level. For a two-tailed test, z* is typically 1.96.
4. Step 4: Calculate the margin of error (E). The margin of error is given by the formula: $E=\mathrm{z*}\times \sqrt{\frac{\mathrm{p̂}}{\mathrm{p̂}}}$. Substitute the values of z*, p̂, and n into the formula.
5. Step 5: Construct the confidence interval. The confidence interval is given by: $[\mathrm{p̂}-E,\mathrm{p̂}+E]$. Substitute the values of p̂ and E to find the interval. Then, compare the interval to the hypothesized proportion (0.512) to determine if the sample results provide strong evidence against the belief.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true proportion. This concept is crucial for estimating population parameters based on sample data.

Proportion

Proportion refers to the fraction of a whole that possesses a certain characteristic, often expressed as a percentage. In this context, it represents the ratio of boys born to the total number of births in the sample. Understanding proportions is essential for interpreting the results of the sample and for constructing the confidence interval around the estimated proportion of boys in the population.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, it involves testing the null hypothesis that the true proportion of boys is 0.512 against the alternative hypothesis that it is different. The results from the confidence interval can help determine whether the sample provides strong evidence against the null hypothesis, guiding conclusions about the population.