Inference about a Population Mean: One-Sample t-Test

1. Chapter 16 Inference about a Population Mean Chapter 16

2. Conditions for Inferenceabout a Mean • Data are a SRS of size n • Population has a Normal distribution with mean m and standard deviation s • Both m and s are unknown • Because s is unknown, we cannot use z procedures to infer µ • This is a more realistic situation than in prior chapters Chapter 16

3. Standard Error • When we do not know population standard deviation s, we use sample standard deviation s to calculate the standard deviation of x-bar, which is now equal to This statistic is called the standard error of the mean Chapter 16

4. One-Sample t Statistic • When we use s instead of s, our “one-sample z statistic” becomes a one-sample t statistic • The t test statistic does NOT follow a Normal distribution  it follows a t distribution with n – 1 degrees of freedom Chapter 16

5. The t Distributions • The t density curve is similar to the standard Normal curve (symmetrical, mean = 0, bell-shaped), but has broader tails • The t distribution is a family of distributions • Each family member is identified by its degree of freedom • Notation t(k) means “a t distribution with k degrees of freedom” Chapter 16

6. The t Distributions As k increases, the t(k) density curve approaches the Z curve more closely. This is because s becomes a better estimate of sas the sample size increases Chapter 16

7. Using Table C Table C gives t critical values having upper tail probability p along with corresponding confidence level C The bottom row of the table applies to z* because a t* with infinite degrees of freedom is a standard Normal (Z) variable Chapter 16

8. Using Table C This t table below highlights the t* critical value with probability 0.025 to its right for the t(7) curve t* = 2.365 (for 7 df) Chapter 16

9. One-Sample t Confidence Interval Take a SRS of size n from a population with unknown mean m and unknown standard deviation s. A level C confidence interval for m is given by: where t* is the critical value for confidence level C from the t(n – 1) density curve. Chapter 16

10. A study of 8 American adults from a 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 (m) is: Illustrative example American Adult Heights “We are 95% confident that the average height of all American adults is between 63.9 and 70.5 inches.” Chapter 16

11. One-Sample t Test The t test is similar in form to the z test learned earlier. The test statistic is: The P-value of t is determined from the t(n – 1) curve via the t table. Chapter 16

12. P-value for Testing Means • Ha: m > m0 • P-value is the probability of getting a value as large or larger than the observed test statistic (t) value. • Ha: m < m0 • P-value is the probability of getting a value as small or smaller than the observed test statistic (t) value. • Ha: m ¹ m0 • P-value is two times theprobability of getting a value as large or larger than the absolute value of the observed test statistic (t) value. Chapter 16

13. Chapter 16

14. Illustrative example Weight Gain To study weight gain, ten people between the age of 30 and 40 determine their change in weight over a 1-year interval. Here are their weight changes: 2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3 Are these data statistically significant evidence that people in this age range gain weight over a given year? Chapter 16

15. Weight Gain Illustration Hypotheses: H0: m = 0 Ha: m > 0 Test Statistic:(df = 101 = 9) P-value:P-value = P(T > 2.70) = 0.0123 (using a computer)P-value is between 0.01 and 0.02 since t = 2.70 is between t* = 2.398 (p = 0.02) and t* = 2.821 (p = 0.01) (Table C) Conclusion:Since the P-value is smaller than a = 0.05, there is significant evidence that against the null hypothesis. Chapter 16

16. Illustrative example Weight Gain Chapter 16

17. Matched Pairs t Procedures • Matched pair samples allow us to compare responses to two treatments in paired couples • Apply the one-sample t procedures to the observed differences within pairs • The parameter m is the mean difference in the responses to the two treatments within matched pairs in the entire population Chapter 16

18. These 8 differences have = 1.0113 and s = 0.1960. Case Study – Matched Pairs Air Pollution • Pollution index measurements were recorded for two areas of a city on each of 8 days • To analyze, subtract Area B levels from Area A levels. The 8 differences form a single sample. • Are the average pollution levels the same for the two areas of the city? Chapter 16

19. Case Study Hypotheses: H0: m = 0 Ha: m ≠ 0 Test Statistic:(df = 81 = 7) P-value:P-value = 2P(T > 14.594) = 0.0000017 (using a computer)P-value is smaller than 2(0.0005) = 0.0010 since t = 14.594 is greater than t* = 5.408 (upper tail area = 0.0005) (Table C) Conclusion:Since the P-value is smaller than a = 0.001, there is strong evidence (“highly signifcant”) that the mean pollution levels are different for the two areas of the city. Chapter 16

20. Case Study Air Pollution 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 0.8474 to 1.1752 index points. Chapter 16

21. Valid data? Unconfounded comparison? (lurking variables are balanced) Simple random sample? Normal distribution? Conditions for t procedures Weight Gain Illustration Except in the case of small samples, conditions 1 –3 are of more importance than condition 4 (t procedures are “robust” to violations in condition 4). Chapter 16

22. Robustness of Normality Assumption • The t confidence interval and test are exact when the distribution of the population is Normal • However, no real data are exactly Normal • A confidence interval or significance test is robust when the confidence level or P-value does not change very much when the conditions for use of the procedure are violated • t statistics are robust when n is moderate to large Chapter 16

23. t Procedure Robustness • Sample size less than 15: Use t procedures if the data appear about Normal (symmetric, single peak, no outliers). If the data are skewed or if outliers are present, do not use t. • Sample size at least 15: The t procedures can be used except in the presence of outliers or strong skewness in the data. • Large samples: The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ≥ 40. Chapter 16

24. 0 6 8 9 1 1 1 1 2 2 Assessing Normality Air Pollution Illustration • Recall the air pollution illustration • Is it reasonable to assume the sampling distribution of xbar is Normal? • While we cannot judge Normality from just 8 observations, a stemplot of the data (right) shows no outliers, clusters, or extreme skewness. • Thus, the t test will be accurate. Chapter 16

25. Can we use t? The stemplot shows a moderately sized data set (n = 20) with a strong negative skew. We cannot trust t procedures in this instance Chapter 16

26. Can we use t? • This histogram shows the distribution of word lengths in Shakespeare’s plays. The sample is very large. • The data has a strong positive skew, but there are no outliers. We can use the t procedures since n ≥ 40 Chapter 16

27. Can we use t? • This histogram shows the heights of college students. • The distribution is close to Normal, so we can trust the t procedures for any sample size. Chapter 16