Question

Part 2

Part 1

Answer 1

1. (a)

As it is very much evident, this is a do by yourself exercise. That is, you need to run the simulations using the Applet provided to you. So the test statistics of the confidence interval must be generated at your end.

However, I can guide about what are the expected values and the methodology to obtain them, and what is the theory that works behind confidence intervals, so that you can corroborate your working against the actual findings.

So as we can see here, we need to do two kinds of tests:

first, we need to vary the confidence level itself, viz. the alpha value, by keeping sample size and proportion constant.

Secondly, we need to vary the sample size, keeping the confidence level and proportion constant. Let us see the theory behind it.

for given proportion p and sample size n, the standard error of proportion is defined as

SE = $\sqrt{p(1-p)/n}$.

The confidence interval for a given level, $\alpha$ is always centered about the given proportion itself. From Z-table, the two-sided critical value, $Z_(\alpha/2)$.

Then, C.I. of proportion is

In expanded form,

The above formulae completely describes the dependence of confidence interval on the confidence level $\alpha$, and the sample size n. For the first part of activity, note that the Z-value increases as alpha decreases, i.e. test becomes more significant. For ex, a 99% confidence level ($\alpha$ = 0.01) will give us a much wider interval than a 95% confidence interval.

A caveat here is that do not assume a linear relationship between the confidence level and interval width - Pick up the Z-table, and read the appropriate critical values. The Z-value for 95% confidence is 1.96, whereas for 99% confidence is 2.34. All other factors remaining constant, we conclude that length of C.I. increases by about 19%.

Now, it has been asked here that do we obtain the same proportion of intervals each time we run the proportion test. The idea here is that the above formulae will give us theoretical values. But in actual test scenario when we run the simulation, the values obtained are expected to be "reasonably" close to the theoretical values, but we should not expect 100% conformance.

Hence, the answer here is that we shall expect slight deviation, perhaps single digit percentage points, which can be considered within tolerable bounds, and reflects that our theoretical values are a good model for the problem.

For the second part of activity, we are asked to vary the sample size. From the formula, we note that C.I. is inversely proportional to square root of n. Hence, the relationship is inverse, though not in the first degree.

So the C.I. for n=40, against n=10 will be halved, and for n=100 will be reduced to one-tenth. As sample size increases, the confidence interval narrows down to a much smaller range, because in a larger sample the distribution of proportion values attains a higher peak about the mean, viz. smaller variance.

This leads to a much smaller confidence interval. Similar to the previous activity, here also the actual values will only expected to be in the vicinity of theoretical values.

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