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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 of this type lands point up is between 0.502 and 0.698.

Step by step solution

01

Identify Necessary Values

The necessary values are the number of 'successes' (x) and the total number of trials (n). Here, a success is when a tack lands point up. According to the exercise, x is 60 (the number of tacks that landed point up) and n is 100 (total number of tacks thrown).
02

Calculate the Sample Proportion

The sample proportion (\(p\)) is the ratio of successes to the number of trials, \(p = \frac{x}{n} = \frac{60}{100} = 0.6. This represents our best estimate for the probability of a tack landing point up.
03

Apply the Confidence Interval Formula

The formula for a 95% confidence interval around a proportion is given by \[p \pm z\sqrt{\frac{p(1-p)}{n}}\] where \(p\) is the sample proportion and \(z\) is the Z-score corresponding to the desired level of confidence (1.96 for 95% confidence level). Inputting the respective values we get: \[0.6 \pm 1.96 \sqrt{\frac{0.6 \times 0.4}{100}}\].
04

Compute the Confidence Interval

Doing the above algebra gives us \(0.6 \pm 0.098\), which means that our 95% confidence interval for the probability of a tack landing point up is between 0.502 and 0.698.

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