Ratio of the distance of two percentiles

Question

Consider two sample means $\bar{X_1}$ and $\bar{X_2}$ from the same normal population having mean $\mu$ and standard deviation $\sigma$. The first sample mean is based on $n_1 = 10^k$ observations, while the second is based on $n_2 = 10^{k+2}$ observation for some positive integer $k$. Take any percentile, say $100 * \alpha$, from the distribution of the means for each sample size excluding the median. What is the ratio of the distance of the two percentiles from $\mu$ dividing distance 1 by distance 2?

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I need help breaking this question down. I understand that we initially have two sample means from the same population. I understand that both samples are based on different sample sizes involving 10 raised to some power of k for the first sample, and k+1 for the second.

But I start to lose the thread after this. Why does the instruction take pains to "exclude the median"? Is this important, or is this just a red herring? Are we now considering two samples of sample means, with each sample either belonging to a collection of samples of size $10^k$ and $10^{k+1}$? And how would I even begin to measure the distance from the mean? Does this involve standardizing the percentile to a z-score?

Answer 1

To "help breaking this question down" as you asked, here are some general steps:

1. Choose any $\alpha$ which is different than 50%, e.g. $\alpha=0.05$. This is important because your population is normal, hence the (population) mean is equal to the (population) median and the distance would be zero if you were to use $\alpha=0.5$.
2. You can calculate the percentiles $p_1$ and $p_2$ using a formula which involves the parameters of your distribution, $\alpha$, and the sample sizes $n_1$ and $n_2$ respectively. (Think of what the percentiles represent, and that should help you choose the apropriate function to use for this step.)

Editing to add some sub-steps for step 2:

• First, write what the distributions of $\bar X_1$ and $\bar X_2$ are;
• You will find that the variances differ (intuitively, since the second sample has more data, you anticipate that the variance for $\bar X_2$ will be smaller);
• Then, write the formula for the percentiles for each sample; although these are "ugly", most of this will simplify in the next step.

3. Calculating the ratio of the distances should then be fairly straightforward (because when subtracting $\mu$, most of the remaining terms cancel out): you should be left with a constant, due to the fact that n2/n1 is a constant.