1 Happiness comes not from material wealth but less desire.

Chapter 17 Inference about a Population Mean BPS - 5th Ed.Chapter 172

Conditions for Inference about a Mean BPS - 5th Ed.Chapter 173  Data are from a SRS of size n.  Population has a Normal distribution with mean  and standard deviation .  Both  and  are usually unknown.  we use inference to estimate .  Problem:  unknown means we cannot use the z procedures previously learned.

Standard Error BPS - 5th Ed.Chapter 174  When we do not know the population standard deviation  (which is usually the case), we must estimate it with the sample standard deviation s.  When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic.  The standard error of the sample mean is

One-Sample t Statistic BPS - 5th Ed.Chapter 175  When we estimate  with s, our one-sample z statistic becomes a one-sample t statistic.  By changing the denominator to be the standard error, our statistic no longer follows a Normal distribution. The t test statistic follows a t distribution with n – 1 degrees of freedom.

The t Distributions BPS - 5th Ed.Chapter 176  The t density curve is similar in shape to the standard Normal curve. They are both symmetric about 0 and bell-shaped.  The spread of the t distributions is a bit greater than that of the standard Normal curve (i.e., the t curve is slightly “fatter”).  As the degrees of freedom increase, the t density curve approaches the N(0, 1) curve more closely. This is because s estimates  more accurately as the sample size increases.

The t Distributions BPS - 5th Ed.Chapter 177

Using Table C BPS - 5th Ed.Chapter 178  Table C on page 693 gives critical values having upper tail probability p along with corresponding confidence level C.  z* values are also displayed at the bottom.

Using Table C BPS - 5th Ed.Chapter 179  Find the value t* with probability to its right under the t(7) density curve. t* = 2.365

One-Sample t Confidence Interval BPS - 5th Ed.Chapter 1710 Take an SRS of size n from a population with unknown mean  and unknown standard deviation . A level C confidence interval for  is: –This interval is exact when the population distribution is Normal and approximate for large n in other cases. where t* is the critical value for confidence level C from the t density curve with n – 1 degrees of freedom.

Case Study BPS - 5th Ed.Chapter 1711 American Adult Heights A study of 7 American adults from an SRS yields an average height of = 67.2 inches and a standard deviation of s = 3.9 inches. A 95% confidence interval for the average height of all American adults (  ) is: “We are 95% confident that the average height of all American adults is between and inches.”

One-Sample t Test BPS - 5th Ed.Chapter 1712 Like the confidence interval, the t test is close in form to the z test learned earlier. When estimating  with s, the test statistic becomes: –The P-value is exact when the population distribution is Normal and approximate for large n in other cases. where t follows the t density curve with n – 1 degrees of freedom, and the P-value of t is determined from that curve.

P -value for Testing Means BPS - 5th Ed.Chapter 1713  H a :  >  0 v P-value is the probability of getting a value as large or larger than the observed test statistic (t) value.  H a :  <  0 v P-value is the probability of getting a value as small or smaller than the observed test statistic (t) value.  H a :  0 v P-value is two times the probability of getting a value as large or larger than the absolute value of the observed test statistic (t) value.

BPS - 5th Ed.Chapter 1714

Case Study BPS - 5th Ed.Chapter 1715 Sweetening Colas (Ch. 14) Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by 10 tasters for a new cola recipe: Are these data good evidence that the cola lost sweetness during storage?

Case Study BPS - 5th Ed.Chapter 1716 It is reasonable to regard these 10 carefully trained tasters as an SRS from the population of all trained tasters. While we cannot judge Normality from just 10 observations, a stemplot of the data shows no outliers, clusters, or extreme skewness. Thus, P-values for the t test will be reasonably accurate. Sweetening Colas

Case Study BPS - 5th Ed.Chapter Hypotheses:H 0 :  = 0 H a :  > 0 2. Test Statistic: (df = 10  1 = 9) 3. P-value: P-value = P(T > 2.70) = (using a computer) P-value is between 0.01 and 0.02 since t = 2.70 is between t* = (p = 0.02) and t* = (p = 0.01) (Table C) 4. Conclusion: Since the P-value is smaller than  = 0.02, there is quite strong evidence that the new cola loses sweetness on average during storage at room temperature.

Case Study BPS - 5th Ed.Chapter 1718 Sweetening Colas

BPS - 5th Ed.Chapter 1719 Matched Pairs t Procedures  To compare two treatments, subjects are matched in pairs and each treatment is given to one subject in each pair.  Before-and-after observations on the same subjects also calls for using matched pairs.  To compare the responses to the two treatments in a matched pairs design, apply the one-sample t procedures to the observed differences (one treatment observation minus the other).  The parameter  is the mean difference in the responses to the two treatments within matched pairs of subjects in the entire population.

BPS - 5th Ed.Chapter 1720 Air Pollution Case Study Pollution index measurements were recorded for two areas of a city on each of 8 days. Are the average pollution levels the same for the two areas of the city? Area AArea BA – B

BPS - 5th Ed.Chapter 1721 Air Pollution Case Study It is reasonable to regard these 8 measurement pairs as an SRS from the population of all paired measurements. While we cannot judge Normality from just 8 observations, a stemplot of the data shows no outliers, clusters, or extreme skewness. Thus, P-values for the t test will be reasonably accurate These 8 differences have = and s =

BPS - 5th Ed.Chapter Hypotheses:H 0 :  = 0 H a :  ≠ 0 2. Test Statistic: (df = 8  1 = 7) 3. P-value: P-value = 2P(T > ) = (using a computer) P-value is smaller than 2(0.0005) = since t = is greater than t* = (upper tail area = ) (Table C) 4. Conclusion: Since the P-value is smaller than  = 0.001, there is very strong evidence that the mean pollution levels are different for the two areas of the city. Case Study

BPS - 5th Ed.Chapter 1723 Find a 95% confidence interval to estimate the difference in pollution indexes (A – B) between the two areas of the city. (df = 8  1 = 7 for t*) We are 95% confident that the pollution index in area A exceeds that of area B by an average of to index points. Case Study Air Pollution

Chapter 18 Two-Sample Problems BPS - 5th Ed.Chapter 1824

Two-Sample Problems BPS - 5th Ed.Chapter 1825  The goal of inference is to compare the responses to two treatments or to compare the characteristics of two populations.  We have a separate sample from each treatment or each population.  Each sample is separate. The units are not matched, and the samples can be of differing sizes.

Case Study BPS - 5th Ed.Chapter 1826 Exercise and Pulse Rates A study if performed to compare the mean resting pulse rate of adult subjects who regularly exercise to the mean resting pulse rate of those who do not regularly exercise. This is an example of when to use the two-sample t procedures. nmeanstd. dev. Exercisers Nonexercisers

Conditions for Comparing Two Means BPS - 5th Ed.Chapter 1827  We have two independent SRSs, from two distinct populations  that is, one sample has no influence on the other-- matching violates independence  we measure the same variable for both samples.  Both populations are Normally distributed  the means and standard deviations of the populations are unknown  in practice, it is enough that the distributions have similar shapes and that the data have no strong outliers.

Two-Sample t Procedures BPS - 5th Ed.Chapter 1828  In order to perform inference on the difference of two means (  1 –  2 ), we’ll need the standard deviation of the observed difference :

Two-Sample t Confidence Interval BPS - 5th Ed.Chapter 1829  Draw an SRS of size n 1 form a Normal population with unknown mean  1, and draw an independent SRS of size n 2 form another Normal population with unknown mean  2.  A confidence interval for  1 –  2 is: –here t* is the critical value for confidence level C for the t density curve. The degrees of freedom are equal to the smaller of n 1 – 1 and n 2 – 1.

Case Study BPS - 5th Ed.Chapter 1830 Exercise and Pulse Rates Find a 95% confidence interval for the difference in population means (nonexercisers minus exercisers). “We are 95% confident that the difference in mean resting pulse rates (nonexercisers minus exercisers) is between 4.35 and beats per minute.”

Two-Sample t Significance Tests BPS - 5th Ed.Chapter 1831  Draw an SRS of size n 1 form a Normal population with unknown mean  1, and draw an independent SRS of size n 2 form another Normal population with unknown mean  2.  To test the hypothesis H 0 :  1 =  2, the test statistic is: u Use P-values for the t density curve. The degrees of freedom are equal to the smaller of n 1 – 1 and n 2 – 1.

P -value for Testing Two Means BPS - 5th Ed.Chapter 1832  H a :  1  >  2 v P-value is the probability of getting a value as large or larger than the observed test statistic (t) value.  H a :  1  <  2 v P-value is the probability of getting a value as small or smaller than the observed test statistic (t) value.  H a :  1  2 v P-value is two times the probability of getting a value as large or larger than the absolute value of the observed test statistic (t) value.

Case Study BPS - 5th Ed.Chapter 1833 Exercise and Pulse Rates Is the mean resting pulse rate of adult subjects who regularly exercise different from the mean resting pulse rate of those who do not regularly exercise?  Null: The mean resting pulse rate of adult subjects who regularly exercise is the same as the mean resting pulse rate of those who do not regularly exercise? [H 0 :  1 =  2 ]  Alt: The mean resting pulse rate of adult subjects who regularly exercise is different from the mean resting pulse rate of those who do not regularly exercise? [H a :  1 ≠  2 ] Degrees of freedom = 28 (smaller of 31 – 1 and 29 – 1).

Case Study BPS - 5th Ed.Chapter Hypotheses:H 0 :   =  2 H a :   ≠   2. Test Statistic: 3. P-value: P-value = 2P(T > 3.961) = (using a computer) P-value is smaller than 2(0.0005) = since t = is greater than t* = (upper tail area = ) (Table C) 4. Conclusion: Since the P-value is smaller than  = 0.001, there is very strong evidence that the mean resting pulse rates are different for the two populations (nonexercisers and exercisers).

Avoid Inference About Standard Deviations BPS - 5th Ed.Chapter 1835  There are methods for inference about the standard deviations of Normal populations.  Most software packages have methods for comparing the standard deviations.  However, these methods are extremely sensitive to non-Normal distributions and this lack of robustness does not improve in large samples.  Hence it is not recommended that one do inference about population standard deviations in basic statistical practice.