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Chapter 4: Problem 20

When 100 tacks were thrown on a table, 60 of them landed point up. Obtain a \(95 \%\) confidence interval for the probability that a tack of this type lands point up. Assume independence.

Short Answer

Expert verified
The 95% confidence interval for the probability that a tack lands point up is approximately (0.504, 0.696).

Step by step solution

01

Derive Parameters

The sample proportion \(\hat{p}\) is derived from the given values. It is the number of successful outcomes divided by the total number of trials: \(\hat{p} = \frac{60}{100} = 0.6\). The sample size \(n\) is 100.
02

Find Z-Score

Look up the corresponding Z-score for a 95% confidence level in a Z-table or use a calculator. The Z-score for a 95% confidence interval is approximately 1.96.
03

Calculate the Standard Error

Compute the standard error using the formula \(SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\). Substituting the known values gives us: \(SE = \sqrt{\frac{0.6(1 - 0.6)}{100}} = 0.049\
04

Calculate Confidence Interval

Compute the confidence interval using the formula \(CI = \hat{p} \pm z * SE\). Substituting the known values gives us: \(CI = 0.6 \pm 1.96 * 0.049\), which gives us the interval (0.504, 0.696).

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