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Many homeowners buy detectors to check for the invisible gas radon in their homes. We want to determine the accuracy of these detectors. University researchers placed 12 radon detectors in a chamber that exposed them to 102 picocuries per liter of radon. The detector readings were: 91.9, 103.8, 97.8, 99.6, 111.4, 96.6, 122.3, 119.3, 105.4, 104.8, 95.0, 101.7. Assume that the standard deviation (\(\sigma\)) is 9 for the population of all radon detectors. We want to determine if there is convincing evidence at the 90% confidence level that the mean reading of all detectors of this type differs from the true value of 105.

Our hypotheses are:
- Null: \(\mu = 105\)
- Alternative: \(\mu \neq 105\)

A significance test was performed, resulting in a test statistic of \(z = -0.3336\) and a P-value of 0.74.

1. Describe what a Type I error would be in this situation.

2. Calculate the probability of a Type I error in this situation.

3. Describe what a Type II error would be in this situation.

Answer :

A type 1 error in this situation would be rejecting the null hypothesis that the mean reading of all detectors of this type is equal to 105 when, in reality, it is true.

In hypothesis testing, a type 1 error occurs when we reject the null hypothesis, which assumes no significant difference or effect, when it is actually true. In this specific scenario, the null hypothesis states that the mean reading of all detectors of this type is 105. If we commit a type 1 error, it means we incorrectly conclude that there is evidence to suggest that the mean reading differs from 105, even though it does not.

A type 1 error is essentially a false positive, where we mistakenly detect an effect or difference that doesn't exist. In the context of this study, it would mean falsely concluding that the detectors are inaccurate in measuring radon levels, despite there being no convincing evidence to support this claim.

Learn more about hypothesis:

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