Bus 621 Statistics Lecture 1 Basics of Statistical

Bus 621 Statistics Lecture 1 Basics of Statistical Inference

Lecture 1 1. Inference for a single numerical variable 2. Statement of hypotheses 3. P-value concept 4. How to communicate the results of a test 5. Inference for a single numerical variable and a categorical variable with 2 categories 6. Inference for a single categorical variable 7. Inference for 2 categorical variables

Statistical Methods Descriptive Statistics Inferential Statistics Tutorials Estimation Hypothesis Testing

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

Unknown Population Parameters Are Estimated Estimate Population Parameter. . . Mean with Sample Statistic x Proportion p Std. Dev. s 1 - 2 x 1 - x 2 Differences p

Estimation Methods Estimation Point Estimation Interval Estimation

Point Estimation 1. Provides a single value • Based on observations from one sample 2. Gives no information about how close the value is to the unknown population parameter 3. Example: Sample mean x = 3 is a point estimate of unknown population mean

Interval Estimation 1. Provides a range of values • Based on observations from one sample 2. Gives information about closeness to unknown population parameter 3. Example: Unknown population mean lies between 50 and 70 with 95% confidence

Confidence Level 1. Probability that the unknown population parameter falls within interval 2. Denoted (1 – • is probability that parameter is not within interval 3. Typical values are 99%, 95%, 90%

Intervals & Confidence Level Sampling Distribution of Sample Mean _ /2 1 - /2 x = _ X (1 – α)% of intervals contain μ α% do not Large number of intervals

Factors Affecting Interval Width 1. Data dispersion More variability = larger width 2. Sample size Larger sample = smaller width 3. Level of confidence (1 – ) Higher confidence = larger width © 1984 -1994 T/Maker Co.

Accurate Confidence Interval for Mean ( Unknown) Assumption: Population must be normally distributed

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, 3. 1. What is the 90% confidence interval estimate of the population mean task time?

Confidence Interval for a Mean ( ) with Unknown , Using Mega. Stat • Mega. Stat does all calculations for you. We can be 90% confident that the population mean falls between 3. 379 and 4. 021.

Applications An example using L 1 One sample numerical variable. xlsx Problem 1: Obtain and interpret a 95% confidence interval for the population mean for price per square foot for all combinations of SAD and with/without a pool. Problem 2: Check to see if these confidence intervals may be inaccurate by looking at normality/sample size. Your Turn: Do PS 1 problem 1

Statistical Methods Descriptive Statistics Inferential Statistics Estimation Hypothesis Testing

What’s a Hypothesis? A belief about a population parameter I believe the mean GPA of this class is 3. 5! • Parameter is population mean, proportion, slope • Must be stated before analysis © 1984 -1994 T/Maker Co.

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

How do we Measure “Close”? 1. 2. 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 the p-value is smaller than, say, 5%, we “reject” the hypothesized value for .

Basic Idea Sampling Distribution It is unlikely that we would get a sample mean of this value. . . therefore, we reject the hypothesis that = 50. . if in fact this were the population mean 20 = 50 H 0 Sample Means

Naming Null & Alternative Hypotheses 1. Null hypothesis, H 0 (pronounced H-oh) always has equality sign: , , or 2. Alternative hypothesis, Ha , opposite of null 3. Ha always has inequality sign: , , or 4. Specified as Ha : , , 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 Ha

What Are the Hypotheses? Is the population average amount of TV viewing equal to 12 hours? • State the question statistically: = 12 • State the opposite statistically: 12 • Select the alternative hypothesis: Ha: 12 • State the null hypothesis: H 0: = 12 • This is a two-tailed test.

What Are the Hypotheses? 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: H 0: = 12 • This is a two-tailed test.

What Are the Hypotheses? 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: H 0: 25 • This is a one-tailed or right-tailed test.

What Are the Hypotheses? 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: H 0: ≥ 20 • This is a one-tailed or left-tailed test.

Level of Significance 1. 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. 2. Designated (alpha) • Typical values are. 01, . 05, . 10 (. 05 is most common) 3. Selected by researcher, otherwise will be given in a problem 4. Defines unlikely values of sample statistic if null hypothesis is true

p-Value Approach 1. Probability of obtaining a test statistic more extreme ( or than actual sample value, given H 0 is true is called the p-value 2. 1 - (p-value) is called the confidence in Ha 3. 1 - is called the required confidence to conclude Ha 4. Used to make a decision between hypotheses • If confidence in Ha is greater than the required confidence, conclude Ha otherwise find H 0 acceptable.

The Four Steps of a Hypothesis Test 1. State Hypotheses 2. Determine p-value (Mega. Stat) 3. Make decision based on 1 -p =confidence in Ha 4. Draw conclusion within context of problem • If confidence in Ha is greater than the required confidence, conclude Ha otherwise find H 0 acceptable.

t Test for Mean ( Unknown) Assumption for p-value to be accurate • Population is normally distributed • If not normal, take large sample (n 30) • Or switch to a test for population median such as Wilcoxon Mann-Whitney test

One-Tailed t Test Example Is the average capacity of batteries less than 140 ampere-hours? A random sample of 20 batteries had a mean of 138. 47 and a standard deviation of 2. 66. Assume a normal distribution. Test at the. 05 level of significance.

One-Tailed t Test Solution • • H 0: Ha: = df = ≥ 140 < 140. 05 20 - 1 = 19 p-value =. 009 (Mega. Stat) We can be 99. 1% confident that Conclusion: the population mean is less than 140 and since that exceeds the requirement of 95% we can conclude < 140

One-Tailed t Test You’re a marketing analyst for Wal. Mart. Wal-Mart had teddy bears on sale last week. The weekly sales ($00 s) of bears sold in 10 stores was: 8 11 0 4 7 8 10 5 8 3 At the. 05 level of significance, is there evidence that the average bear sales per store is more than 5 ($00 s)?

One-Tailed t Test Solution* • • • H 0: 5 p-value =. 111 from Mega. Stat Ha: > 5 Confidence in Ha = 1 -. 111 or =. 05. 889 df = 10 - 1 = 9 Required confidence to There is insufficient evidence conclude Ha is 95%. that pop. mean is more than 5 since we can be only 88. 9% confident.

One-tailed T-test for a Mean ( ) with Unknown , Using Mega. Stat

One-tailed T-test for a Mean ( ) with Unknown , Using Mega. Stat 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 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 3. 238 lb. with a standard deviation of. 117 lb. At the. 01 level of significance, is the manufacturer correct? 3. 25 lb.

Two-Tailed t Test Solution* • • • H 0: = 3. 25 p-value =. 208 from Mega. Stat Confidence in Ha = 1 -. 208 or. 792 Ha: 3. 25 . 01 df 64 - 1 = 63 Need to be 99% confident to conclude Ha There is insufficient evidence pop. mean is not 3. 25 since we can only be 79. 2% confident. The null hypothesis is acceptable.

Applications An example using L 1 One sample numerical variable. xlsx Problem 3: Test the hypothesis that the mean price per square foot mean for SAD 3 Pool is different than $320 at a level of significance of. 05. How does that compare to the 95% confidence interval you calculated in Problem 1. Use a level of significance of. 05 in this problem and all that follow. Problem 4: Use the Wilcoxon signed rank test to test whether the median price per square foot for SAD 2 Pool is different than $320. Problem 5: Test the hypothesis that the mean price per square foot for SAD 1 No. Pool is less than 350. Example 6: Use the Wilcoxon signed rank test to test whether the median price per square foot for SAD 1 No. Pool is less than $350. Your Turn: Do PS 1 problems 2, 3, 4

Two Independent Populations Example applications 1. An economist wishes to determine whethere is a difference in mean family income for households in two socioeconomic groups. 2. 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. How can we tell what to use for these situations? • Both have a numerical variable and a categorical variable (with 2 categories) • See “Choosing Situation by Data Type”

Comparing Two Independent Means, μ 1 – μ 2, assuming unknown Assumptions • Independent, random samples • Populations are approximately normally distributed • Population standard deviations are equal If at least one population is not normal then an alternative test is to compare population medians using the Wilcoxon Mann-Whitney test

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 11 15 Mean 3. 27 2. 53 Std Dev 1. 30 1. 16 Assuming normal populations, is there a difference in average yield ( =. 05)? © 1984 -1994 T/Maker Co.

Independent Samples Hypothesis Test Solution • • • . p-value =. 1397 H 0: 1 - 2 = 0 ( 1 = 2) Confidence in Ha Ha: 1 - 2 0 ( 1 2) . 05 Is 1 -. 1397 =. 8603 df 11 + 15 - 2 = 24 Need to be 95% confident to conclude Ha There is little evidence of a difference in means since we can only be 86. 03% confident that the pop. means are different

Two Sample T-test & C. I. for Mean Difference Assuming Equal Variances, Using Mega. Stat

Two Sample T-test & C. I. for Mean Difference Assuming Equal Variances, Using Mega. Stat Hypothesis Test: Independent Groups (t-test, pooled variance) NYSE 3. 27 1. 3 11 NASDAQ 2. 53 mean 1. 16 std. dev. 15 n 24 df 0. 740 difference (NYSE - NASDAQ) 1. 489 pooled variance 1. 220 pooled std. dev. 0. 484 standard error of difference 0 hypothesized difference 1. 53 t. 1397 p-value (two-tailed) -0. 260 confidence interval 95. % lower 1. 740 confidence interval 95. % upper 1. 000 margin of error

Wilcoxon Mann-Whitney test using Mega. Stat Wilcoxon - Mann/Whitney Test n 32 29 61 Pr/SF sum of ranks 1035. 5 SAD 1 Pool 855. 5 SAD 2 Pool 1891 total expected 992. 000 value standard 69. 243 deviation z corrected for ties with continuity 0. 621 correction. 5346 p-value (two-tailed) H 0: Population Medians are equal H 1: Population Medians are not equal P-value =. 5346 We can only be 46. 54 % confident of a difference in population medians. See L 1 2 sample tests excel file for this example.

Applications An example using the L 1 2 sample tests. xlsx excel file. Example 7: Test whether price per square foot has the same population means for homes with and without pools. Your Turn: Do PS 1 problems 5, 6, 7

A single categorical variable: Z Test for a Proportion 1. Condition • nπ and n(1 -π) > 5 2. Z-test from Mega. Stat Example: Do ranch style homes make up less than 50% of the population of homes? Data: A sample of 108 homes revealed that 54 were ranch style.

One-Tailed Solution • H 0: π ≥ 0. 50 • Ha: π < 0. 50 • • =. 05 P-value =. 0271 from Excel Mega. Stat We can be 97. 29% confident that the population proportion is less than 0. 5 and therefore can conclude that π < 0. 50 95% confidence interval estimate for π We can be 95% confident that the population proportion falls between. 3147 and. 5001. Note: A 2 -tailed test would have found the null hypothesis acceptable.

Applications An example using the L 1 Categorical variables tests and CI-1. xls file. Example 8: Test whether less than 50% of the homes are ranch style in the population and obtaining a 95% interval estimate for that population proportion. Your Turn: Do PS 1 problems 8

Two categorical variables: Chi-square Test for Independence 1. Chi-square test statistic Example: Do 3 different school districts have the same percentage of ranch, trilevel and two-story homes? Data: A sample of 108 homes revealed the following table. Count of STYLE Grand SAD ranch trilevel twostory Total SAD 1 8 24 11 SAD 2 15 11 7 SAD 3 21 4 7 Grand Total 44 39 25 43 33 32 108

Chi-square solution • H 0: No relationship • Ha: Relationship exists • • =. 05 P-value =. 0005 from Excel Mega. Stat We can be 99. 95% confident that there is a relationship between school district and style of home A follow up analysis suggests that SAD 1 has fewer ranch homes and more trilevel homes than expected and that the reverse holds for SAD 3. See the Results tab in L 1 Categorical variables file for details.

Applications An example using the L 1 Categorical variables tests and CI-1. xlsx file. Example 9: Test whethere is a relationship between SAD and style of home. Your Turn: Do PS 1 problems 9