Making Decisions about a Population Mean with Confidence Lecture 35 Sections 10.1 – 10.2 Fri, Mar 25, 2005

Introduction In Chapter 10 we will ask the same basic questions as in Chapter 9, except they will concern the mean. In Chapter 10 we will ask the same basic questions as in Chapter 9, except they will concern the mean. Find an estimate for the mean. Find an estimate for the mean. Test a hypothesis about the mean. Test a hypothesis about the mean.

The Steps of Testing a Hypothesis (p-Value Approach) 1. State the null and alternative hypotheses. 1. State the null and alternative hypotheses. 2. State the significance level. 2. State the significance level. 3. Compute the value of the test statistic. 3. Compute the value of the test statistic. 4. Compute the p-value. 4. Compute the p-value. 5. State the conclusion. 5. State the conclusion.

The Hypotheses The null and altenative hypotheses will be statements concerning . The null and altenative hypotheses will be statements concerning . Null hypothesis. Null hypothesis. H 0 :  =  0. H 0 :  =  0. Alternative hypothesis (choose one). Alternative hypothesis (choose one). H 1 :    0. H 1 :    0. H 1 :  <  0. H 1 :  <  0. H 1 :  >  0. H 1 :  >  0.

Level of Significance The level of significance is the same as before. The level of significance is the same as before. If the value is not given, assume that  = If the value is not given, assume that  = 0.05.

The Test Statistic The choice of test statistic will depend on the sample size and what is known about the population. The choice of test statistic will depend on the sample size and what is known about the population. For the moment, we will assume that  is known for the population. For the moment, we will assume that  is known for the population. Later we will consider the case where  is unknown. Later we will consider the case where  is unknown.

The Sampling Distribution of  x If the population is normal, then the distribution of  x is also normal, with mean  0 and standard deviation  /  n. If the population is normal, then the distribution of  x is also normal, with mean  0 and standard deviation  /  n. That is,  x is N(  0,  /  n). That is,  x is N(  0,  /  n). Note that this assumes that  is known. Note that this assumes that  is known. See p. 500, the Central Limit Theorem. See p. 500, the Central Limit Theorem.

The Sampling Distribution of  x Therefore, the test statistic is Therefore, the test statistic is It is exactly standard normal. It is exactly standard normal.

The Sampling Distribution of  x On the other hand, if On the other hand, if the population is not normal, the population is not normal, but the sample size is at least 30, but the sample size is at least 30, then the distribution of  x is approximately normal, with mean  0 and standard deviation  /  n. That is,  x is approximately N(  0,  /  n). That is,  x is approximately N(  0,  /  n). Note that we are still assuming that  is known. Note that we are still assuming that  is known. See p. 500, the Central Limit Theorem. See p. 500, the Central Limit Theorem.

The Sampling Distribution of  x Therefore, the test statistic is Therefore, the test statistic is It is approximately standard normal. It is approximately standard normal. The approximation is good enough that we can use the normal tables. The approximation is good enough that we can use the normal tables.

Decision Tree Is  known? yesno

Decision Tree Is  known? yesno Is the population normal? yesno

Decision Tree Is  known? yesno Is the population normal? yesno

Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno

Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno

Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up

Decision Tree Is  known? yesno Is the population normal? yesno Is n  30? yesno Give up TBA

Example See Example 10.1, p. 569 – Too Much Carbon Monoxide? (  known). See Example 10.1, p. 569 – Too Much Carbon Monoxide? (  known).

Let’s Do It! Let’s Do It! 10.2, p. 573 – Completing a Maze. Let’s Do It! 10.2, p. 573 – Completing a Maze.

Hypothesis Testing on the TI-83 Press STAT. Press STAT. Select TESTS. Select TESTS. Select Z-Test. Press ENTER. Select Z-Test. Press ENTER. A window appears requesting information. A window appears requesting information. Select Data if you have the sample data entered into a list. Select Data if you have the sample data entered into a list. Otherwise, select Stats. Otherwise, select Stats.

Hypothesis Testing on the TI-83 Assuming you selected Stats, Assuming you selected Stats, Enter  0, the hypothetical mean. Enter  0, the hypothetical mean. Enter . (Remember,  is known.) Enter . (Remember,  is known.) Enter  x. Enter  x. Enter n, the sample size. Enter n, the sample size. Select the type of alternative hypothesis. Select the type of alternative hypothesis. Select Calculate and press ENTER. Select Calculate and press ENTER.

Hypothesis Testing on the TI-83 A window appears with the following information. A window appears with the following information. The title “Z-Test.” The title “Z-Test.” The alternative hypothesis. The alternative hypothesis. The value of the test statistic Z. The value of the test statistic Z. The p-value of the test. The p-value of the test. The sample mean. The sample mean. The sample size. The sample size.

Example Re-do Example 10.1 on the TI-83 (using Stats). Re-do Example 10.1 on the TI-83 (using Stats). The TI-83 reports that The TI-83 reports that z = – z = – p-value = p-value =

Hypothesis Testing on the TI-83 Suppose we had selected Data instead of Stats. Suppose we had selected Data instead of Stats. Then somewhat different information is requested. Then somewhat different information is requested. Enter the hypothetical mean. Enter the hypothetical mean. Enter . Enter . Identify the list that contains the data. Identify the list that contains the data. Skip Freq (it should be 1). Skip Freq (it should be 1). Select the alternative hypothesis. Select the alternative hypothesis. Select Calculate and press ENTER. Select Calculate and press ENTER.

Hypothesis Testing on the TI-83 Why enter  if the TI-83 is capable of computing the standard deviation from the data? Why enter  if the TI-83 is capable of computing the standard deviation from the data?

Example Re-do Example 10.1 on the TI-83 (using Data). Re-do Example 10.1 on the TI-83 (using Data). Enter the data in the chart on page 570 into list L 1. Enter the data in the chart on page 570 into list L 1. The TI-83 reports that The TI-83 reports that z = z = p-value = p-value =  x =  x = s = (  4.8). s = (  4.8).

Let’s Do It! Re-do Let’s Do It! 10.2, p. 573, using the TI-83. Re-do Let’s Do It! 10.2, p. 573, using the TI-83.