Elementary Statistics—Inference for a Population Mean

(Out of our bubble! We no longer assume we know the population standard deviation.)

Setting

Suppose we have a random sample of size n from a normal population with unknown mean, , and unknown standard deviation. (Note: Since the population standard deviation is unknown, we must use the t-distribution, not the z-distribution, for inference.)

Confidence Interval

A level C confidence interval for is

• The value comes from the T-distribution (Table D) with (n – 1) degrees of freedom. It’s the value such that there’s area C between - and . (Now you get to use “magical” Table D to find these values—no reverse lookup.)

• Recall s is simply our notation for the sample standard deviation (as defined in Section 1.2).

• Per usual, our confidence is in the method we use, not in our one particular interval.
• The quantity is often called the standard error (just another name for standard deviation) of the sample mean.
• The quantity is still called the margin of error.

Significance Test

Suppose we want to test . To test this hypothesis, first calculate the test statistic: [Note this is simply a standardized value. But since we estimated the population standard deviation with the sample standard deviation, we must look up the P-value in the T-table, not the z-table.]

Then determine the P-value using the T-distribution with (n – 1) degrees of freedom. Recall that the P-value depends on the direction of the alternative hypothesis (e.g., if the alternative hypothesis is two-sided, then you need to double the P-value).

Finally, define the P-value in the words of the problem and provide a conclusion (which might depend on a given value of significance, ). If you find statistical significance, then it’s a good idea to create a confidence interval to assess the practical significance.

Important Notes:

• The t-inference procedures require that the population being sampled follows a normal distribution. We can check this condition by looking at a graph of the sample data. Also, as n gets larger, we can relax the condition of normality of the population (since the sampling distribution of is more normal—by the CLT—and since s gets closer to ). Here is our rule of thumb for checking the normality condition of the t-test (depending on the sample size):
• If , then we must be very careful and only use the t-procedures if the sample-data distribution is close to normal (which indicates the population distribution is close to normal);

• If , then we have more “wiggle room” and we can use the t-procedures even if our sample-data distribution is somewhat skewed (but we shouldn’t use the procedures if the sample-data distribution is strongly skewed or has extreme outliers);

• If , then we have lots of “wiggle room” and we can use the t-procedures even if our sample-data distribution is strongly skewed (but we shouldn’t use the procedures if there are extreme outliers).

• General note: Regardless of the sample size, if the sample-data distribution looks fairly normal, then the normality condition on the population has been met.

• For a given set of data, if it’s inappropriate (based on the above rule-of-thumb) to use the t-procedures, then there are other statistical options (e.g., non-parametric tests, bootstrap procedures); we just don’t have time to discuss these procedures this term (but you can take Math 217: Applied Statistical Methods if you want to learn more!).