**Example**

A study was conducted to determine the mean amount that hair grows daily. A sample of 35 young adults (from 18 to 35 years of age) was randomly selected from all patrons at a large hair salon. The amount of daily hair growth was recorded for each person in the sample. The sample mean was 0.35 mm, and the sample standard deviation was 0.05 mm. Set a 90% confidence interval on the mean daily hair growth of young adults. Interpret the interval in the context of the problem.

**Solution**

First, we need to determine whether the conditions for inference are satisfied. The first condition that the sample be a random one is satisfied. Although we are not told that the population distribution of daily hair growth is normally distributed, this would seem to be a reasonable assumption. If the data were available, we could construct a histogram and a boxplot to determine whether the assumption of normality is reasonable. However, because *n* = 35 > 30, we can appeal to the Central Limit Theorem and assume that the sampling distribution of is approximately normal.

Because the standard deviation is unknown, we use the following confidence interval:

We are given that *n* = 35, = 0.35 mm, and *s* = 0.05 mm. The *t** value we need would have n –1 = 34 degrees of freedom and = = 0.025 probability in the upper tail. Looking in the *t*-table in Lesson 12, we have *t** = 2.032. The confidence limits are 0.35 ± 2.032 or 0.35 ± 0.02. That is, we are 95% confident that the mean daily hair growth of young adults who are patrons of this salon is between 0.33 and 0.37 mm. Note: Because the population of young adult patrons of this salon was the one from which the sample was selected, it is the one to which we can draw inference. Does the inferene extend to all young adults at least in some region about the salon? It does if the patrons of the salon do not differ significantly from others in the region with respect to hair growth. Careful thought should be given before inferences are extended to a broader population than that sampled.

**Hypothesis Testing for a Mean**

As with proportions, there are five steps to conducting a test of hypotheses. We will consider each separately.

**Step 1: Specifying the Hypotheses**

Remember that, when establishing a set (*H*_{0} and *H _{a}*) of statistical hypotheses, the research hypothesis is the alternative hypothesis. When working with means, the null hypothesis is that the population mean μ is equal to some value μ

_{0}The alternative may be that μ is less than, greater than, or equal to μ

_{0},depending on what the research hypothesis is.

**Step 2: Verify Necessary Conditions for a Test and, if Satisfied, Construct the Test Statistic**

The conditions for testing hypotheses about the population mean μ are the same as those for constructing a confidence interval on this parameter. If the population distribution is normal, it is enough to know that we have a random sample from the population distribution. If the population distribution is not normal, in addition to having a random sample from the population, the sample size must be large enough to assume that the sampling distribution of is approximately normal (i.e., *n* ≥ 30).

If the population is normal and the standard deviation is unknown or if the population distribution is not known but a large sample (*n* ≥ 30) has been selected, the test statistics is where μ_{0} is the value of the mean under the null hypothesis.

Notice that the first test statistics has the form .

If the standard deviation of the population and hence the standard deviation of the point estimate are known, we certainly want to use the true value of the parameter instead of the standard error, its estimate. This would lead us to use a different test statistic from that presented here. The reality is that, in practice, the population is rarely, if ever, known, and we will only consider this case.

**Step 3: Find the p-Value Associated with the Test Statistic**

If the population distribution is normal, the test statistic has a *t*-distribution with (*n* – 1) degrees of freedom. Similarly, if the population distribution is not normal but a large sample has been selected, the test statistic has an approximate *t*-distribution. If the null hypothesis is not true, the test statistic does not have a *t*-distribution. The observed value of the test statistic *t _{T}* is likely to be unusual for a randomly selected observation from the

*t*-distribution. The

*p*-value continues to be the probability of observing a value as extreme as or more extreme than the test statistic,

*t*, from a random selection of an observation from a

_{T}*t*-distribution with (

*n*– 1) degrees of freedom.

How do we measure how unusual a test statistic is? It depends on the alternative hypothesis. These are summarized in Table 17.1.

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