Help with confidence intervals (mean)

Question

A researcher is interested in finding a 90% confidence interval for the mean number of times per day that college students text. The study included 106 students who averaged 38.3 texts per day. The standard deviation was 19.9 texts. Round answers to 3 decimal places where possible.

With 90% confidence the population mean number of texts per day is between  ? and  ? texts.

If many groups of 106 randomly selected members are studied, then a different confidence interval would be produced from each group. About ? percent of these confidence intervals will contain the true population number of texts per day and about ? percent will not contain the true population mean number of texts per day. 

Answer 1

I'll break this down into two separate portions based on the question above (as it seems to be a two part question). The first part involves calculating the interval itself.

Part A:

To calculate a confidence interval for a sample, we use the formula: CI = sample mean ± (Z * s/√n) where:

CI = Confidence Interval

Z = critical value (found by looking up the alpha value, which I'll cover shortly)

s = standard deviation of the sample

n = number of observations in sample

The alpha value is found by subtracting the desired confidence interval % as a decimal (in this case, 90% = 0.9) from 1. Then, we have to determine if we're looking at a one- or two-tailed confidence interval. Since we're looking at an interval that encapsulates the mean, it'll be a two-tailed test. So we need to use an alpha value of 0.05 for each tail. (In other words, when we look up the critical value, we are going to be using that critical value to calculate how far from the mean the left and right values are. If we use the full 0.1 alpha value instead of 0.05, we'll be calculating a confidence interval of 10% each direction, and collectively, we'll have accidentally calculated our interval to have an 80% confidence interval because we've doubled the original 10%. Thus, we cut the alpha value in half when we're looking at two-tailed confidence intervals so that, when we combine the two "tails", we get the original alpha value of 10%, or 0.1.)

There are very common alpha values that are regularly used; a 90% confidence interval has alpha values of 0.1 for one-tailed tests and 0.05 for two-tailed tests. When you look up critical value tables for z-values, you'll find that the critical value for this alpha level is 1.64. Thus, z=1.64 in the calculation above.

From there, you can easily calculate the remainder of the equation by plugging in the values provided in the original problem. Just make sure you round to 3 decimal places!

Part B:

This is just making sure that you understand the definition of a confidence interval. Whenever we calculate a confidence interval, we set the original % confidence we want to end up with before we even start calculating. This % actually impacts the calculation based on the alpha value that was discussed above; the higher the percentage desired, the smaller the alpha value (and thus the larger the critical number).

So when we calculate the confidence interval, we're actually calculating a range of values that we're 90% (or whatever percent we set originally) confident that the target value is contained within. That means that if we were to re-calculate that interval 100 times, we'd expect 90 of those intervals to contain the true population mean while there would be about 10 of those that don't contain the true population mean.

Hopefully this helps!