Unit 7. Day 16.

Some statistical questions we might ask: On average, do boys and girls differ on quantitative reasoning? Do students learn basic arithmetic skills better with or without calculators? Which of two medications is more effective in treating migraine headaches? Does one type of car get better mileage per gallon of gasoline than another type? Does one type of fabric decay in landfills faster than another type? Do people with diabetes heal more slowly than people who do not have diabetes? Two Populations

Comparing Boys & Girls Heavier Taller Younger Stupider Lighter Shorter Older Smarter

Example A: Tamika’s mathematics project is to see whether boys or girls are faster in solving a KenKen-type puzzle. She creates a puzzle and records the following times that it took to solve the puzzle (in seconds) for a random sample of 10 boys from her school and a random sample of 11 girls from her school:   Mean MAD Boys 39 38 27 36 40 43 34 33 35.3 4.04 Girls 41 42 47 32 46 39.4 3.96

Faster Slower ?

Example A: Draw dot plots for the boys’ data and for the girls’ data   Mean MAD Boys 39 38 27 36 40 43 34 33 35.3 4.04 Girls 41 42 47 32 46 39.4 3.96 Comment on the amount of overlap between the two dot plots. How are the dot plots the same, and how are they different? The dot plots appear to have a considerable amount of overlap. The boys’ data may be slightly skewed to the left, whereas the girls’ are relatively symmetric

Example A: Draw dot plots for the boys’ data and for the girls’ data   Mean MAD Boys 39 38 27 36 40 43 34 33 35.3 4.04 Girls 41 42 47 32 46 39.4 3.96 The variability in each data set is about the same as measured by the mean absolute deviation (around 𝟒 𝒔𝒆𝒄.) For boys and girls, a typical deviation from their respective mean times (𝟑𝟓 for boys and 𝟑𝟗 for girls) is about 𝟒 𝒔𝒆𝒄. Compare the variability in the two data sets using the MAD (mean absolute deviation). Is the variability in each sample about the same? Interpret the MAD in the context of the problem.

Difference of the means: 39.4−35.3 = 4.1 4.1 Since the difference in the sample means is nearly equal to the MAD, we conclude that the population means won’t be significantly different. Example A: Draw dot plots for the boys’ data and for the girls’ data   Mean MAD Boys 39 38 27 36 40 43 34 33 35.3 4.04 Girls 41 42 47 32 46 39.4 3.96 4.04 Difference of the means: 39.4−35.3 = 4.1 4.1 𝑣𝑠.

The Difference in our Sample Means is not “Meaningful.” Faster Slower The Difference in our Sample Means is not “Meaningful.”

Number of Minutes of Texting Example B: Suppose that Brett randomly sampled 12 tenth-grade girls and boys in his school district and asked them for the number of minutes per day that they text. The data and summary measures follow. There is no overlap between the two data sets. This indicates that the sample means probably differ, with girls texting more than boys on average. The girls’ data set is a little more compact than the boys, indicating that their measure of variability is smaller Draw dot plots for the two data sets using the same numerical scales. Discuss the amount of overlap between the two dot plots that you drew and what it may mean in the context of the problem. Gender Number of Minutes of Texting   Mean MAD Girls 98 104 95 101 107 86 92 96 88 97.3 5.3 Boys 66 72 65 60 78 82 63 56 85 79 68 77 70.9 7.9

Text less Text more ?

Number of Minutes of Texting Example B: The MAD for the boys’ number of minutes spent texting is 𝟕.𝟗 𝐦𝐢𝐧., which is higher than that for the girls, which is 𝟓.𝟑 𝐦𝐢𝐧. This is not surprising, as seen in the dot plots. The typical deviation from the mean of 𝟕𝟎.𝟗 is about 𝟕.𝟗 𝐦𝐢𝐧. for boys. The typical deviation from the mean of 𝟗𝟕.𝟑 is about 𝟓.𝟑 𝐦𝐢𝐧. for girls. Compare the variability in the two data sets using the MAD. Interpret the result in the context of the problem. Gender Number of Minutes of Texting   Mean MAD Girls 98 104 95 101 107 86 92 96 88 97.3 5.3 Boys 66 72 65 60 78 82 63 56 85 79 68 77 70.9 7.9

Does the difference in the two means appear to be meaningful? Since the difference in the sample means is more than three times the MAD, we conclude that the population means will be significantly different. Example B: Using the larger MAD of 𝟕.𝟗 𝐦𝐢𝐧., the means are separated by 𝟑.𝟑 MADs. Looking at the dot plots, it certainly seems as though a separation of more than 𝟑 MADs is meaningful Does the difference in the two means appear to be meaningful? Gender Number of Minutes of Texting   Mean MAD Girls 98 104 95 101 107 86 92 96 88 97.3 5.3 Boys 66 72 65 60 78 82 63 56 85 79 68 77 70.9 7.9 7.9 Difference of the means: 97.3−70.9 = 26.4 26.4 𝑣𝑠.

The Difference in our Sample Means IS “Meaningful.” Text less Text more The Difference in our Sample Means IS “Meaningful.”

The Balcony (1868) Homework 7.16: No Homework