Confidence Intervals and Tests of Hypotheses for Means Study Guide (page 2)
Introduction to Confidence Intervals and Tests of Hypotheses for Means
The basic ideas of confidence intervals and hypothesis testing were used to develop large sample confidence intervals and hypothesis tests for population proportions. Here, we will learn how these ideas apply when we are interested in the population mean. Because rarely, if ever, is the population standard deviation known when the population mean is unknown, we will consider only the case where both are unknown.
Confidence Intervals for a Mean
Suppose we have a random sample from a distribution with unknown mean and standard deviation. Further, the distribution may or may not be normal. We would estimate the population mean using the point estimate . How would we set the confidence interval on the population mean? First, we need to determine whether the conditions for the statistical methods used here have been met.
To use the statistical methods we will present here to set a confidence interval on the population mean, two conditions must be satisfied for the methods to be valid. (1) The sample must be randomly selected from the population. (2) The population distribution must be normal, or the sample size must be large enough (at least 30) to assume the sampling distribution of is approximately normal by the Central Limit Theorem.
We learned that the general form of a confidence interval is point estimate ± multiplier × standard error, and this is the form we will use. The point estimate of the population mean is , and the standard error of is . Because t* with (n – 1) degrees of freedom is the multiplier. Thus, the form of the confidence interval for a sample mean is:
if the population distribution is normal with known standard deviation. The choice of t* depends on the confidence level chosen, just as z* did when determining the multiplier for proportions. For a 100 (1 – α)% confidence interval, t* is the t-value for which P(t > t*) = , where t is a random selection from a t-distribution with (n –1) degrees of freedom. (By having in each tail, a total of a is in the tails, leaving 1 – α a for the confidence level. See Figure 17.1.)
A particular species of finch exhibits a polymorphism in bill size that is unrelated to gender. There is a large-billed morph and a small-billed morph. The bill widths within a morph are normally distributed. A random sample of 20 finches is selected from a 100- hectare study region. The average bill width of the sampled finches is 15.92, and the sample standard deviation is 0.066 millimeter. Set a 95% confidence interval on the mean bill width of this species of finch within the study region.
Because the sample was selected randomly from a normal distribution, the conditions for inference are satisfied. The point estimate of the mean bill width is = 15.92. We are given that s = 0.066 and n = 20, so the standard error of is = = 0.0148. For a 95% confidence interval, the t-value corresponding to a t with n – 1 = 19 degrees of freedom and = 0.025 probability in each tail is found by looking in the t-table in Sampling Distributions and the t-Distribution. The column for 0.025 probability in the upper tail and the column for 19 degrees of freedom intersect to give t* = 2.093.Thus, the confidence interval limits are 15.92 ± 2.093 × or 15.92 ± 0.03.We are 95% confident that the mean bill length of the finch species is within 0.03 mm of 15.92 mm in the study region.
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.
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 (H0 and Ha) 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 μ0 The alternative may be that μ is less than, greater than, or equal to μ0,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 μ0 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 tT 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, tT, from a random selection of an observation from a 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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