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Statistical Analysis and Hypothesis Testing: A Comprehensive Study of t-tests, ANOVA, and Probability Calculations

This comprehensive study covers various statistical analyses including t-tests and ANOVA methods.

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    Statistical Analysis and Hypothesis Testing: A Comprehensive Study of t - tests, ANOVA, and Probability Calculations 1. What is the probability of rolling a four in the gambling dice game of craps (given two six sided dice)? There are 3 possibilities of rolling a 4 (3+1, 2+2, 1+3) and a total of 36 possible outcomes. The probability of rolling a 4 is 3/36 = 1/12 = 0.083 What is the probability that a player can roll a four 3 times in a row (assume that rolling the dice each time does not affect the outcome of the next roll)? .083 x .083 x .083 = 0.00058 2 . Population A and Population B both have a mean height of 70.0 inches with an SD of 6.0. A random sample of 30 people is picked from population A, and random sample of 50 people is selected from Population B. Which sample mean will probably yield a more accurate e stimate of its population mean? Why? SEM(A): 6/SQRT(30) = 1.095 SEM(B): 6/SQRT(50) = 0.848 Sample B will yield the more accurate population mean because the SEM decreases as the population increases, meaning that we achieve a closer mean of the overall mean. 3 . Suppose we obtained data on vein size after application of a nitroglycerin ointment in a sample of 5 0 patients. The mean vein size is found to be 8.7 mm with an SD of 2. 1 . Using a t distribution table, what are the confidence limits for a 95% confidence interval? For a 99% confidence interval? SEM 2.1/SQRT(50) = 0.2969 SEM = .2969 95% 99% T - dist. 2.01 2.66 CI 8.7 mm +/ - (2.01 X 0.2969) 8.7 mm +/ - (2.66 X 0.2969) UCL= 9.29mm, LCL= 8.10 mm UCL = 9.49mm, LCL =7.91 mm 4. In a pilot study evaluating the use of a new drug to lower resting heart rates (HR) of patients, the following data was recorded: Subject # Resting HR 001 72 002 88 003 71 004 87 005 64 006 77 007 79 008 59 009 66

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    010 68 011 78 012 89 013 91 014 81 015 77 016 75 017 69 Given that the average resting HR of the general population for this study is 72, use StatCrunch to perform the appropriate t test. a) What is the value of t ? The Null hypothesis states that the mean of the sample is 72 with the only difference being a random variation. The alternative hypothesis states that the mean of the sample is NOT 72 and did not occur in a random way. The calculated t - statistics is 1.75 The critical table value for t is 2.12 b) Using an alpha of 0.05, is the t statistic significant? Why? My calculated t - statistic is less than the table critical value of 2.12. The results are not statistically significant because the t value is smaller than the critical table value of t. Also, the p - value is greater than the alpha value of 0.05. Therefore, we fail to reject the null hypothesis because the results probably did occur by chance. c) What are the confidence limits for a 95% confidence interval here and what do th ey mean for this patient group? Copy and Paste your work from StatCrunch into your Word document submission. The confidence limits for a 95% confidence interval are: 71.15 < x < 80.73. This is , without considering the population mean of 72 , as indicat ed in the problem. This means that we are 95 percent confident that the interval from 71.15 and 80.73 does contain the true value of the population mean. Hypothesis test results: μ : Mean of variable H 0 : μ = 72 H A : μ ≠ 72 Variable Sample Mean Std. Err. DF T - Stat P - value HR 75.941176 2.2581073 16 1.7453451 0.1001 95% confidence interval results: μ : Mean of variable Variable Sample Mean Std. Err. DF L. Limit U. Limit HR 75.941176 2.2581073 16 71.154203 80.72815 5 . Write one or two sentences that c ould be used to report the results obtained for the t - test in Exercise 4 . The experiment in Exercise 4 ended up being not statistically significant. According to the null hypothesis the mean of the population equals to 72 and the results occurred in a random way. On the other hand, the alternative hypothesis tells us that the mean of the population is not 72 and

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