homework_7PBHL-B302

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Indiana University, Purdue University, Indianapolis *

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Statistics

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Apr 3, 2024

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PBHL-B302 Assignment #7 Note: For all hypothesis testing problems, please make sure that you write the hypotheses, test statistic, p-value, and conclusions in terms of the problem. 1. For each of the following situations, identify the response variable and the populations to be compared, and give I ,n i , and N . (1) (5 points) A developer of a virtual-reality (VR) teaching tool for the deaf wants to compare the effectiveness of different navigation methods. A total of 40 children were available for the experiment, of which equal numbers were randomly assigned to use a joystick, wand, dance mat, or gesture-based pinch gloves. The time (in seconds) to complete a designed VR path is recorded for each child. The response variable is time to complete VR path. The population is the 40 children. There are 4 groups. There will be 10 kids per group. There are 40 kids in total. (2) (5 points) To study the effects of pesticides on birds, an experimenter randomly (and equally) allocated 65 chicks to five diets (a control and four with a different pesticide included). After a month, the calcium content (milligrams) in a 1-centimeter length of bone from each chick was measured. The response variable is calcium content. The population are the 65 chicks. There are 5 groups, 4 tests and a control. There are 13 chicks per group. There are 65 chicks in total. 2. To compare the means of 5 populations, random samples of size 10 are selected from each population. A partial ANOVA table for this data is as follows: Source df SS MS F P-value Between Groups 4 39.2192 9.8048 <0.0001 Within Groups 45 28.8 Total 49 68.019 a. (5 points) Fill in the missing entries in the ANOVA table. b. (2 points) Find the pooled estimate of the population standard deviation. c. (1 point) State the null and alternative hypotheses in the experiment. Null: The mean for each populations are the same Alternative: Not all means are equal d. (2 points) Test the hypotheses in part c). Use α=0.05.

Does exercise frequency have an association with pulse rate? 110 students in an introductory statistics course were classified into three groups according to their frequency of exercises: 1 = high frequency, 2 = moderate frequency, 3 = low frequency. Their pulse rates were measured. The data are in “ pulse.csv ”. 3. (2 points) Use software to make a side-by-side boxplot of the three exercise frequency groups, and describe what you find. Remember to discuss whether there are outliers, the amount of overlap between the boxes, and the general trends present. Exercise 1 Exercise 2 Exercise 3 Based off the box plots, exercise 2 and 3 each have outliers, 2 and 1 respectively, and the heart rate of the exercises decreases with each exercise, as exercise 1 is the lowest mean and exercise 3 is the highest mean. The overlap between is there in some of the numbers, more so between 2 and 3 rather than them and 1.

4. (2 points) Use software to find the sample size, sample mean, and sample standard deviation of each exercise frequency group. 5. (2 points) Use software to create a means plot. Discuss the results and how this compares to the side-by-side boxplot in #1. Based on this, the comparison is almost the same, as 1 still has the lowest mean and exercise 3 has the highest mean 6. (2 points) Is it reasonable to pool the variances for one-way ANOVA? Why or why not? Yes, as the smallest standard deviation 9.81 * 2 is greater than the biggest standard deviation of 11.45 7. (2 points) State your hypotheses for a one-way ANOVA test for this story. Null: μ 1 = μ 2 = μ3 Alternative: At least one is different

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8. (2 points) Run the one-way ANOVA in your preferred software, regardless of your answer to #4. Report the F test statistic and the P-value. F = 5.126 P-value = .0075 9. (2 points) What are your conclusions in terms of the pulse rate story (using a significance level of 0.05)? Due to the p-value being under .05, that means that the null hypothesis can be rejected, meaning that at least 1 pulse is different from the others. 10. (2 points) Report the R-squared and the pooled estimate of the standard deviation. R 2 = .087 or 8.7% s p = 109.6 11. (2 points) Is it necessary to do the Bonferroni’s multiple comparison procedure? Give your reason why or why not. It is necessary to do Bonferroni’s multiple comparison procedure as it only works on datasets that have rejected the null hypothesis, and in this case, this dataset rejected the null hypothesis.

12. (2 points) Regardless of your answer to #9, run the Bonferroni procedure using the SPSS output below and report your results in terms of the story. Based on the comparisons from the procedure Multiple Comparisons Dependent Variable: pulse rate Bonferroni -6.241 3.150 .150 -13.90 1.42 -10.423* 3.325 .007 -18.51 -2.34 6.241 3.150 .150 -1.42 13.90 -4.182 2.222 .188 -9.59 1.22 10.423* 3.325 .007 2.34 18.51 4.182 2.222 .188 -1.22 9.59 (J) exercise frequency moderate low high low high moderate (I) exercise frequency high moderate low Mean Difference (I-J) Std. Error Sig. Lower Bound Upper Bound 95% Confidence Interval The mean difference is significant at the .05 level. *.

According to William Butler Yeats, “She is the Gaelic muse, for she gives inspiration to those she prosecutes. The Gaelic poets die young, for she is restless, and will not let them remain long on earth.” One study designed to investigate this issue examined the age at death for writers from different cultures and genders. Three categories of writers examined were novelists, poets, and nonfiction writers. The ages at death for female writers in these categories from North America are given in the file deadpoets.csv . 3. (2 points) Use the graphical and numerical methods to describe the data. 4. (2 points) Examine the assumptions necessary for ANOVA. Summarize your findings. 5. (4 points) Run the ANOVA and report the results. 6. (2 points) Using the multiple comparisons provided below, compare the three groups of writers. What do you find? Bonferroni (I) Type of writers (J) Type of writers Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound Novelist Poet 8.260 * 3.107 .027 .72 15.80 Nonfiction writer -5.427 3.440 .352 -13.78 2.92 Poet Novelist -8.260 * 3.107 .027 -15.80 -.72 Nonfiction writer -13.688 * 3.904 .002 -23.17 -4.21 Nonfiction writer Novelist 5.427 3.440 .352 -2.92 13.78 Poet 13.688 * 3.904 .002 4.21 23.17 *. The mean difference is significant at the 0.05 level.

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