Interactive Exercises - Chapter 5

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Problem W5.1

Use the widget for the law of large numbers for this exercise with the population parameters \(\mu=3\) and \(\sigma=0.2\).

1. What is the smallest sample size \(n\) that gives a probability of at most \(0.01\) for the event that the sample mean deviates from its mean \(\mu = 3\) by at least \(\epsilon = 0.1\) units? That is, use the widget to determine the smallest \(n\) for which \[\mathrm{Pr}(\vert \bar{X}_n - 3 \vert > 0.1) \leq 0.01.\]

Solution

The sample size \(n=26\) gives \[\mathrm{Pr}(\vert \bar{X}_n - 3 \vert > 0.1) \approx 0.01079\] so this sample size is not large enough. However, for \(n=27\) we get \[\mathrm{Pr}(\vert \bar{X}_n - 3 \vert > 0.1) \approx 0.009375,\] which is smaller than the required probability of \(0.01\). So \(n=27\) is the smallest possible sample size. Check for yourself:

2. Let’s be even more demanding now and require that the sample mean can deviate by at most \(\epsilon = 0.01\) units from the mean \(\mu\). What is now the smallest sample size \(n\) that achieves this?

Solution

The sample size \(n=2654\) is the smallest \(n\) and \[\mathrm{Pr}(\vert \bar{X}_n - 3 \vert > 0.01) \approx 0.009999\]

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Problem W5.2

Use the widget for the central limit theorem for this exercise.

1. Choose the Beta distribution with parameters \(\alpha=0.5\) and \(\beta=0.5\) as the data distribution. Set sample size \(n=2\) and look at the orange histogram that shows the sampling distribution of the sample mean for a sample of size \(n=2\). Does it look normally distributed? Continue to increase sample size \(n\) to 3, 4, 5 and so on. How large \(n\) do you need for the sampling distribution to be approximately normal?

Solution

For \(n=2\) the distribution is no longer bathtub shaped, but it is clearly not normal (yet). It is hard to say exactly of course, but already for \(n=10\) is the sampling distribution roughly bell shaped like the normal distribution.

2. repeat Problem W5.2a, but now for the chi-squared distribution with \(\nu=3\) degrees of freedom.

Solution

It takes at least until \(n=20\) for the sampling distribution to no longer have the long right hand tail of the chi-squared distribution.

3. repeat Problem W5.2a, but now for the Cauchy distribution with location \(m=0\) and scale \(\gamma=1\). How large must \(n\) be before the sampling distribution of \(\bar{X}\) seems to be approximately normal?

Solution

The Cauchy distribution is one of the cases where the central limit theorem does not hold. No matter how large you make \(n\), the distribution of \(\bar{X}\) will never be normal. The mean and variance of the Cauchy do not exist, which violates the assumptions of the theorem; it has so extremely heavy tails that the mean does not exist, even though the Cauchy distribution is symmetric around the location \(m\). 🤯