Question

the effectiveness of a blood-pressure drug is being investigated. an experimenter finds that, on average, the reduction in systolic blood pressure is 83.5 for a sample of size 458 and standard deviation 6.4.
estimate how much the drug will lower a typical patient’s systolic blood pressure (using a 95% confidence level).
enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

answer should be obtained without any preliminary rounding.

Explanation:

Step1: Identify the formula for confidence interval

For a large sample (n > 30), we use the z - distribution. The formula for the confidence interval for the population mean \(\mu\) is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\) (when \(\sigma\) is unknown, we can use \(s\) as an estimate for \(\sigma\) for large samples). Here, \(\bar{x} = 83.5\), \(s=6.4\), \(n = 458\), and for a 95% confidence level, \(z_{\alpha/2}=1.96\) (since the area in the two - tails is \(\alpha=0.05\), so \(\alpha/2 = 0.025\) and \(z_{0.025}=1.96\)).

Step2: Calculate the margin of error (E)

The margin of error \(E=z_{\alpha/2}\frac{s}{\sqrt{n}}\). Substitute the values: \(n = 458\), \(s = 6.4\), \(z_{\alpha/2}=1.96\).
First, calculate \(\sqrt{n}=\sqrt{458}\approx21.4009\). Then \(\frac{s}{\sqrt{n}}=\frac{6.4}{21.4009}\approx0.299\). Then \(E = 1.96\times0.299\approx0.586\).

Step3: Calculate the lower and upper bounds of the confidence interval

The lower bound is \(\bar{x}-E=83.5 - 0.586=82.914\approx82.9\) (rounded to one decimal place).
The upper bound is \(\bar{x}+E=83.5 + 0.586=84.086\approx84.1\) (rounded to one decimal place).

Answer:

\(82.9<\mu<84.1\)