Question

1. a new sneaker claims that it can make male athletes jump higher. a sample of 25 male athletes is asked to jump once with their own sneakers on and once wearing the new sneakers. the average jump increased by 1.5 inches with the new sneakers! assume that σ = 3. find a 95% confidence interval and interpret.

Explanation:

Step1: Find the z - score

For a 95% confidence interval, the z - score \(z_{\alpha/2}\) is 1.96.

Step2: Calculate the margin of error \(E\)

The formula for the margin of error for a population mean when the population standard deviation \(\sigma\) is known is \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\).
Given \(n = 25\), \(\sigma=3\), and \(z_{\alpha/2}=1.96\), we have \(E = 1.96\times\frac{3}{\sqrt{25}}=1.96\times\frac{3}{5}=1.176\).

Step3: Calculate the confidence interval

The formula for the confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu <\bar{x} + E\). Given \(\bar{x}=1.5\), the confidence interval is \(1.5 - 1.176<\mu<1.5 + 1.176\), which simplifies to \(0.324<\mu<2.676\).

Answer:

The 95% confidence interval is \((0.324,2.676)\). This means that we are 95% confident that the True mean increase in jump height for all male athletes when wearing the new sneakers lies between \(0.324\) inches and \(2.676\) inches.