Question

the pulse rates of 144 randomly selected adult males vary from a low of 43 bpm to a high of 123 bpm. find the minimum sample size required to estimate the mean pulse rate of adult males. assume that we want 98% confidence that the sample mean is within 4 bpm of the population mean. complete parts (a) through (c) below.
a. find the sample size using the range rule of thumb to estimate σ.
n = 136
(round up to the nearest whole number as needed.)
b. assume that σ = 12.5 bpm, based on the value s = 12.5 bpm from the sample of 144 male pulse rates.
n = □
(round up to the nearest whole number as needed.)

Explanation:

Step1: Find the z - score for 98% confidence level

For a 98% confidence level, the significance level \(\alpha=1 - 0.98=0.02\). Then \(\alpha/2=0.01\). Using the standard normal distribution table or a calculator, \(z_{\alpha/2}=z_{0.01}\approx 2.33\)

Step2: Use the formula for sample size \(n=(z_{\alpha/2}\sigma/E)^2\)

We are given \(\sigma = 12.5\) bpm and \(E = 4\) bpm. Substitute the values into the formula: \(n=(2.33\times12.5/4)^2\)
First, calculate \(2.33\times12.5 = 29.125\), then \(29.125\div4=7.28125\), and \(n=(7.28125)^2\approx53.0176\)

Answer:

Since we need to round up to the nearest whole number, \(n = 54\)