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Chapter 8: Problem 112

In a study to establish the absolute threshold of hearing, 70 male college freshmen were asked to participate. Each subject was seated in a soundproof room and a \(150 \mathrm{H}\) tone was presented at a large number of stimulus levels in a randomized order. The subject was instructed to press a button if he detected the tone; the experimenter recorded the lowest stimulus level at which the tone was detected. The mean for the group was \(21.6 \mathrm{db}\) with \(s=2.1\). Estimate the mean absolute threshold for all college freshmen and calculate the margin of error.

Short Answer

Expert verified
Based on the sample of 70 male college freshmen, we can estimate that the mean absolute threshold of hearing for all college freshmen is 21.6 dB, with a 95% confidence interval of approximately 21.108 dB to 22.092 dB. The margin of error for our estimate is 0.492 dB, which indicates the degree of uncertainty around the true population mean.

Step by step solution

01

Identify the sample statistics

We are given the sample mean (\(\bar{x}\)) and the sample standard deviation (s) for 70 male college freshmen. The sample mean is \(\bar{x} = 21.6 \mathrm{db}\) and the sample standard deviation is \(s = 2.1\).
02

Calculate the standard error of the sample mean

The standard error of the sample mean (SE) is calculated using the formula: \(SE = \frac{s}{\sqrt{n}}\). Here, \(s\) is the sample standard deviation, and \(n\) is the sample size. In our case, \(s = 2.1\) and \(n = 70\). So, the standard error is: \(SE = \frac{2.1}{\sqrt{70}} \approx 0.251\).
03

Calculate the margin of error

To calculate the margin of error, we need to find the critical value (\(z^*\)) from the standard normal distribution table for a given confidence level (usually 95%). For a 95% confidence level, the critical value is \(z^* = 1.96\). The margin of error (ME) is calculated using the formula: \(ME = z^* \times SE\). In our case, \(z^* = 1.96\) and \(SE = 0.251\). So, the margin of error is: \(ME = 1.96 \times 0.251 \approx 0.492\).
04

Estimate the population mean and calculate the confidence interval

Now, we can estimate the population mean using our sample mean and the margin of error. The 95% confidence interval for the population mean is: \(\bar{x} \pm ME\). In our case, \(\bar{x} = 21.6\) and \(ME = 0.492\). So, the 95% confidence interval for the mean absolute threshold for all college freshmen is: \(21.6 \pm 0.492\), or approximately \((21.108, 22.092) \mathrm{db}\). The estimated mean absolute threshold for all college freshmen is \(21.6 \mathrm{db}\) with a margin of error of \(0.492 \mathrm{db}\).

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