Download Notes on Confidence Intervals and Hypothesis tests | STA 2023 and more Exams Data Analysis & Statistical Methods in PDF only on Docsity! 1 t-distibutions: confidence intervals and hypothesis tests One sample means with unknown sigma 1. A manufacturer of small appliances employs a market research firm to estimate retail sales of its products by gathering information from a sample of retail stores. A random sample of 50 stores in the Midwest sales region were asked to give the number of the manufacturer’s electric can openers that were sold last month. Here are the data: 19 19 16 19 25 26 24 63 22 16 13 26 34 10 48 16 20 14 13 24 34 14 25 16 26 25 25 26 11 79 17 25 18 15 13 35 17 15 21 12 19 20 32 19 24 19 17 41 24 27 a. Display the data with a histogram and boxplot. Describe the distribution. Although the histogram is unimodal, it shows strong skewness to the right with a couple of outliers at the high end. Boxplot reveals three outliers at the high end. b. The distribution of sales is strongly right-skewed because there are many smaller stores and a few very large stores. The use of t procedures here is reasonably safe despite this violation of the normality assumption. Why? Histogram is unimodal which is desirable. Despite strong skewness (according to the histogram) and several outliers (according to the boxplot), t-procedures may be used here because • SRS “A random sample of 50 stores…” • Sample size is 50 which is greater than 40. c. Conduct a test at the 5% level to determine if the mean number of can openers sold last month from all stores in the Midwest sales region is not 19 can openers. 1. 19: 19:0 ≠ = µ µ aH H 2. 05.0=α 3. 56.23=x 523.12≈xS 50=n 2 4. 575.2 1956.23 50 523.12 0 ≈−= − = n Sx x t µ P ( )575.2≥t = tcdf(2.575 , 1E99, 49) ≈ 0.0065 P-value = 2*0.0065 ≈ 0.013 remember two-tailed! Also, STAT →→ TESTS 2 (T-Test) yields same results. 5. P-value α≤ 05.0013.0 ≤ Reject 0H 6. This sample is significant evidence at the 5% level that the mean number of can openers sold last month from all stores in the Midwest sales region is not 19 can openers. d. Give a 95% confidence interval for the mean number of can openers sold by all stores in the region. Interpret the interval. Do 1-VAR-STATS to get 56.23=x and 523.12≈xS . Choose t* from the table in your textbook (the t* table is usually located in the Appendix in the back of the book). The sample size is 50 so it follows that the degrees of freedom df = 49. However, df as 49 does not appear on the table. When this happens it is safer to round down with degrees of freedom on a table, so choose df = 45. Therefore t* = 2.014. This gives us a confidence interval of 50 523.12 014.256.23 ± OR (19.993 , 27.127) Using STAT →→ TESTS 8 (TInterval) on the calculator reveals a more accurate confidence of (20.001 , 27.119) We are 95% confident that the mean number of can openers sold last month from all stores in the Midwest sales region is between 20.001 and 27.119 can openers.
