Learning Objectives Compute Confidence Interval Estimates for Population Mean (Sigma Unknown) Distinguish Types of Hypotheses Describe Hypothesis Testing.

Learning Objectives Compute Confidence Interval Estimates for Population Mean (Sigma Unknown) Distinguish Types of Hypotheses Describe Hypothesis Testing Process Explain p-Value Concept Solve Hypothesis Testing Problems Based on a Single Sample Solve Inference Problems for Two Populations Means As a result of this class, you will be able to ...

Descriptive Statistics Inferential Statistics
Statistical Methods Statistical Methods Descriptive Statistics Inferential Statistics Hypothesis Testing Estimation 5

I am 95% confident that  is between 40 & 60.
Estimation Process Population Mean X= 50 Random Sample  I am 95% confident that  is between 40 & 60. Mean, , is unknown     Sample         7

Estimate Population Parameter...
Unknown Population Parameters Are Estimated Estimate Population Parameter... with Sample Statistic Mean   x Proportion  p Variance  2 s Differences  1 -  2  x -

Estimation Methods Estimation Interval Estimation Point Estimation 14

Point Estimation Provides a single value Based on observations from one sample Gives no information about how close the value is to the unknown population parameter Example: Sample mean x = 3 is point estimate of unknown population mean

Interval Estimation Provides a range of values
Based on observations from one sample Gives information about closeness to unknown population parameter Stated in terms of probability Knowing exact closeness requires knowing unknown population parameter Example: Unknown population mean lies between 50 and 70 with 95% confidence

Confidence limit (lower) Confidence limit (upper)
Key Elements of Interval Estimation Sample statistic (point estimate) Confidence interval Confidence limit (lower) Confidence limit (upper) A probability that the population parameter falls somewhere within the interval. 24

Distribution of Samples Means
 x _ X 90% of means  +1.65x -1.65x 99% of means -2.58x +2.58x 95% of means +1.96x -1.96x  29

Parameter = Statistic ± Error
Confidence Limits for Population Mean Parameter = Statistic ± Error © T/Maker Co. 27

Confidence Level Probability that the unknown population parameter falls within interval Denoted (1 –  is probability that parameter is not within interval Typical values are 99%, 95%, 90%

Sampling Distribution of Sample Mean
Intervals & Confidence Level Sampling Distribution of Sample Mean _  x  /2  /2 1 -  _ X  =   x (1 – α)% of intervals contain μ α% do not Intervals extend from X – ZσX to X + ZσX Large number of intervals

Intervals extend from X – ZX toX + ZX
Factors Affecting Interval Width Data dispersion Measured by  Intervals extend from X – ZX toX + ZX Sample size Have students explain why each of these occurs. Level of confidence can be seen in the sampling distribution. Level of confidence (1 – ) Affects Z © T/Maker Co.

Realistic Confidence Interval for Mean ( Unknown)
Assumptions Population standard deviation is unknown Population must be normally distributed Substitute sample s for the unknown  and use Student’s t–distribution instead of Z distribution

Realistic Confidence Interval for Mean ( Unknown)

Student’s t Distribution
Standard Normal Bell-Shaped Symmetric ‘Fatter’ Tails t (df = 13) t (df = 5) Z t

Degrees of Freedom (df)
Number of observations that are free to vary after sample statistic has been calculated Example Sum of 3 numbers is 6 X = 1 (or any number) X = 2 (or any number) X = 3 (cannot vary) Sum = 6 degrees of freedom = n = = 2

Student’s t Table t  / 2 2.920 t values v t 1 3.078 6.314 2 1.886
Assume: n = 3 df = n - 1 = 2  = .10 /2 =.05 t  / 2 v t .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 3 1.638 2.353 3.182 Confidence intervals use /2, so divide ! t values 2.920 59

Estimation Example Mean ( Unknown)
A random sample of n = 25 has x = 50 and s = 8. Set up a 95% confidence interval estimate for . 70

Thinking Challenge You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, What is the 90% confidence interval estimate of the population mean task time? Allow students about 20 minutes to solve.

Confidence Interval Solution*
x = 3.7 s = n = 6, df = n - 1 = = 5 t.05 = 2.015 72

Confidence Interval for a Mean () with Unknown Using MegaStat
MegaStat give you a choice of z or t and does all calculations for you.

Descriptive Statistics Inferential Statistics
Statistical Methods Statistical Methods Descriptive Statistics Inferential Statistics Hypothesis Testing Estimation 5

I believe the mean GPA of this class is 3.5!
What’s a Hypothesis? A belief about a population parameter Parameter is population mean, proportion, variance Must be stated before analysis I believe the mean GPA of this class is 3.5! © T/Maker Co.

Reject hypothesis! Not close.
Hypothesis Testing I believe the population mean age is 50 (hypothesis). Reject hypothesis! Not close. Population  Mean X = 20 Random sample 

How do we Measure “Close”?
Xbar should not be more than roughly two standard deviations of xbar away from hypothesized value. Why? If hypothesized  value were really the true mean, there should be a high probability of obtaining the observed sample xbar by pure random chance. Call this the p-value If xbar is more than 2*std.devs. away from  or p-value is smaller than, say, 5%, we “reject” the hypothesized value for .

Sampling Distribution
Basic Idea Sample Means  = 50 H0 Sampling Distribution It is unlikely that we would get a sample mean of this value ... 20 ... therefore, we reject the hypothesis that  = 50. ... if in fact this were the population mean

Naming Null & Alternative Hypotheses
Null hypothesis, H0 (pronounced H-oh) always has equality sign: , , or  Specified as H0:   some numeric value Specified with = sign even if  or  Example, H0:   3 instead of  ≤ 3 Alternative hypothesis, Ha or H1, opposite of null Ha (H1) always has inequality sign: ,, or  Specified as Ha (H1):  ,, or  some value Example, Ha:  < 3

Identifying Hypotheses
Example: Test that the population mean is not 3 Steps: State the question statistically (  3) State the opposite statistically ( = 3) Must be mutually exclusive & exhaustive Designate which is alternative hypothesis (  3) Has the , <, or > sign Designate which is the null hypothesis ( = 3) Called a two-tailed hypothesis because of  in H1

What Are the Hypotheses?
Is the population average amount of TV viewing 12 hours? State the question statistically:  = 12 State the opposite statistically:   12 Select the alternative hypothesis: Ha:   12 State the null hypothesis: H0:  = 12 This is a two-tailed test. Why?

Is the population average amount of TV viewing different from 12 hours? State the question statistically:   12 State the opposite statistically:  = 12 Select the alternative hypothesis: Ha:   12 State the null hypothesis: H0:  = 12 This is a two-tailed test. Why?

Is the average amount spent in the bookstore greater than $25? State the question statistically:   25 State the opposite statistically:   25 Select the alternative hypothesis: Ha:   25 State the null hypothesis: H0:   25 This is a one-tailed or right-tailed test. Why?

Is the average cost per hat less than $20? State the question statistically:   20 State the opposite statistically:  ≥ 20 Designate the alternative hypothesis: Ha:   20 State the null hypothesis: H0:  ≥ 20 This is a one-tailed or left-tailed test. Why?

Level of Significance A “tail” probability of the bell curve used to define how many std. devs. of xbar to judge “closeness” and to compare p-value against. Designated (alpha) Typical values are .01, .05, .10 Selected by researcher, otherwise will be given in a problem Defines unlikely values of sample statistic if null hypothesis is true Called rejection region of sampling distribution

Sampling Distribution Observed sample statistic
Rejection Region Ho Value Critical  Sample Statistic Rejection Region Nonrejection Sampling Distribution 1 –  Level of Confidence Observed sample statistic Rejection region does NOT include critical value.

Sampling Distribution
Rejection Region Sampling Distribution Ho Value Critical  Sample Statistic Rejection Region Nonrejection Sampling Distribution 1 –  Level of Confidence Level of Confidence Observed sample statistic Rejection region does NOT include critical value.

p-Value Approach Probability of obtaining a test statistic more extreme (or than actual sample value, given H0 is true Called observed level of significance Smallest value of  for which H0 can be rejected Used to make rejection decision If p-value  , do not reject H0 If p-value < , reject H0

Formal Hypothesis Testing Procedure

The Six Steps of a Hypothesis Test
State Hypotheses Compute Test Statistic (a “t-score”) Set up p-value and critical value decision rules based on significance level  Determine p-value Make decision Draw conclusion within context of problem

t Test for Mean ( Unknown)
Assumptions Population is normally distributed If not normal, only slightly skewed & large sample (n  30) taken t-test statistic

Critical Values of t Table (Portion)
Right or Left -Tailed t Test Finding Critical t Values Given: n = 3;  = .05 v t .10 .05 .025 1 3.078 6.314 12.706 2 1.886 2.920 4.303 3 1.638 2.353 3.182 Critical Values of t Table (Portion)   df = n - 1 = 2  = .05  t 2.920 -2.920 

One-Tailed t Test Example
Is the average capacity of batteries at least 140 ampere-hours? A random sample of 20 batteries had a mean of and a standard deviation of Assume a normal distribution. Test at the .05 level of significance.

One-Tailed t Test Solution
Test Statistic: .005 < p-value <.01 Decision: Conclusion:  ≥ 140  < 140 H0: Ha:  = df = Critical Value: .05 = 19 Reject Ho since t -1.729 .05 Reject H0 p-value <  and |t*| > t-critical There is evidence population average is less than 140

One-Tailed t Test Thinking Challenge
You’re a marketing analyst for Wal-Mart. Wal- Mart had teddy bears on sale last week. The weekly sales ($00s) of bears sold in 10 stores was: At the .05 level of significance, is there evidence that the average bear sales per store is more than 5 ($00s)? Assume that the population is normally distributed. Allow students about 10 minutes to solve this.

One-Tailed t Test Solution*
  5  > 5 Test Statistic: Decision: Conclusion: H0: Ha:  = df = Critical Value: .05 = 9 Note: More than 5 have been sold (6.4), but not enough to be significant. Do not reject Ho t 1.833 .05 Reject H0 p-value > .10>  and |t*| < t-crit There is insufficient evidence average is more than 5

One-tailed T-test for a Mean () with Unknown Using MegaStat

One-tailed T-test for a Mean () with Unknown Using MegaStat
Hypothesis Test: Mean vs. Hypothesized Value 5.0000 hypothesized value 6.4000 mean Sales ($00) 3.3731 std. dev. 1.0667 std. error 10 n 9 df 1.31 t .1109 p-value (one-tailed, upper)

Two-Tailed t Test Example
Does an average box of cereal contain the advertised 368 grams of cereal? A random sample of 36 boxes had a mean of and a standard deviation of 12 grams. Test at the .05 level of significance. Note: t-critical values for two-tailed must be obtained by splitting  into two. 368 gm.

Two-Tailed t Test Solution
Test Statistic: 0.02 < p-value < Decision: Conclusion:  = 368   368 H0: Ha:  = df = Critical Value(s): .05 = 35 Reject Ho since t 2.042 -2.042 .025 Reject H p-value <  = .05 and |t*| > t-crit There is evidence population average is not 368 but is >

Two-Tailed t Test Thinking Challenge
You work for the FTC. A manufacturer of detergent claims that the mean weight of detergent is 3.25 lb. You take a random sample of 64 containers. You calculate the sample average to be lb. with a standard deviation of .117 lb. At the .01 level of significance, is the manufacturer correct? 3.25 lb. Allow students about 10 minutes to finish this.

Two-Tailed t Test Solution*
 = 3.25   3.25 Test Statistic: Decision: Conclusion: H0: Ha:   df  Critical Value(s): .01 = 63 F.T.R. reject Ho since p- t 2.660 -2.660 .005 Reject H value > .10 and |t*| < t-critical There is not enough evidence average is not 3.25

Errors in Making Decisions
Type I Error Reject true null hypothesis Has serious consequences Probability of Type I Error is (alpha) Called level of significance Type II Error Do not reject false null hypothesis Probability of Type II Error is (beta)

Decision Results H0: Innocent Jury Trial H0 Test Actual Situation
True False Accept 1 –  Type II Error () Reject Type I Error () Power (1 – ) Actual Situation Verdict Innocent Guilty Innocent Correct Error Guilty Error Correct

   &  Have an Inverse Relationship
You can’t reduce both errors simultaneously!  

Thinking Challenge How would you try to answer these questions?
Who gets higher grades: males or females? Which program is faster to learn: Word or Excel?

Independent & Related Populations
Different data sources Unrelated Independent Same data source Paired or matched Repeated measures (before/after) Matching Match according to some characteristic of interest. Repeated Measures Assumes the same individual behaves similarly under both treatments except for treatment effect. Any difference will be due to treatment effect. Use difference between the two sample means X1 – X2 Use difference between each pair of observations di = x1i – x2i

Two Independent Populations Examples
An economist wishes to determine whether there is a difference in mean family income for households in two socioeconomic groups. An admissions officer of a small liberal arts college wants to compare the mean SAT scores of applicants educated in rural high schools and in urban high schools. These are comparative studies. The general purpose of comparative studies is to establish similarities or to detect and measure differences between populations. The populations can be (1) existing populations or (2) hypothetical populations.

Two Related Populations Examples
Nike wants to see if there is a difference in durability of two sole materials. One type is placed on one shoe, the other type on the other shoe of the same pair. An analyst for Educational Testing Service wants to compare the mean GMAT scores of students before and after taking a GMAT review course.

Thinking Challenge Are they independent or related?
Miles per gallon ratings of cars before and after mounting radial tires The life expectancy of light bulbs made in two different factories Difference in hardness between two metals: one contains an alloy, one doesn’t Tread life of two different motorcycle tires: one on the front, the other on the back 1. Related. Same car should be used since car weights vary. 2. Independent. Can’t match light bulbs. 3. Independent. No relationship between two alloys. Only differ in alloy. 4. Related. Tires are matched to same motorcycle. Should alternate tire types between front and back and motorcycle.

Comparing Two Independent Means, μ1 – μ2, assuming  unknown
Assumptions Independent, random samples Populations are approximately normally distributed Population variances are equal

Confidence Interval for μ1 – μ2 (Independent Samples)

Confidence Interval Example
You’re a financial analyst for Charles Schwab. You want to estimate the difference in dividend yield between stocks listed on the NYSE and NASDAQ? You collect the following data: NYSE NASDAQ Number Mean Std Dev Assuming normal populations, what is the 95% confidence interval for the difference between the mean dividend yields? © T/Maker Co.

Confidence Interval Solution
df = n1 + n2 – 2 = – 2 = t.025 = 2.064 42

Hypothesis Test for μ1 – μ2 (Independent Samples)
Two Independent Sample t–Test Statistic Hypothesized difference 39

Hypothesis Test Example
You’re a financial analyst for Charles Schwab. Is there a difference in dividend yield between stocks listed on the NYSE and NASDAQ? You collect the following data: NYSE NASDAQ Number Mean Std Dev Assuming normal populations, is there a difference in average yield ( = .05)? © T/Maker Co.

Independent Samples Hypothesis Test Solution
1 - 2 = 0 (1 = 2) 1 - 2  0 (1  2) Test Statistic: Decision: Conclusion: H0: Ha:   df  Critical Value(s): .05 = 24 t 2.064 -2.064 .025 Reject H 41

42

Test Statistic: .1 < p-value < .2 Decision: Conclusion: H0: 1 - 2 = 0 (1 = 2) Ha: 1 - 2  0 (1  2)   .05 df  = 24 Critical Value(s): Do not reject Ho since t 2.064 -2.064 .025 Reject H p-value >  and |t*| < t-crit There is no evidence of a difference in means 41

Relationship between Confidence Interval for μ1 – μ2, and two-tailed Hypothesis Test about difference Assume confidence level (e.g. 95%) and significance level (e.g. 5%) sum to 100% If confidence interval limits go from negative to positive (if interval contains zero) then conclude no difference between the means; do not reject Ho that μ1 – μ2 = 0 If confidence interval is entirely above or below zero conclude Ha: μ1 – μ2 ≠ 0

Relationship between Confidence Interval and two-tailed Hypothesis Test: Charles Schwab Example
95% confidence interval for μ1 – μ2 = to 1.74 Since lower and upper interval limits are negative and positive (interval contains zero) conclude there is no significant difference between μ1 and μ2 Fail to reject Ho: μ1 – μ2 = 0 at  = 5% level Note 5% and 95% sum to 100%

Two Sample T-test & C.I. for Mean Difference Assuming Equal VariancesUsing MegaStat

Learning Objectives Compute Confidence Interval Estimates for Population Mean (Sigma Unknown) Distinguish Types of Hypotheses Describe Hypothesis Testing.

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Learning Objectives Compute Confidence Interval Estimates for Population Mean (Sigma Unknown) Distinguish Types of Hypotheses Describe Hypothesis Testing.

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