BST 322 Week 2 Assignment 1. What Is The Probability Of Roll

BST 322 Week 2 Assignment 1. What Is The Probability Of Rolling A Four

BST 322 Week 2 Assignment 1. What is the probability of rolling a four in the gambling dice game of craps (given two six-sided dice)? 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)?

Population A and Population B both have a mean height of 70.0 inches with a standard deviation of 6.0. A random sample of 30 people is picked from Population A, and a random sample of 50 people is selected from Population B. Which sample mean will probably yield a more accurate estimate of its population mean? Why?

Suppose we obtained data on vein size after application of a nitroglycerin ointment in a sample of 50 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?

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. Given that the average resting HR of the general population for this study is 72, use StatCrunch to perform the appropriate t test. What is the value of t? Using an alpha of 0.05, is the t statistic significant? Why? What are the confidence limits for a 95% confidence interval here and what do they mean for this patient group? Copy and paste your work from StatCrunch into your Word document submission.

Write one or two sentences that could be used to report the results obtained for the t-test in Exercise 4.

For which of the following situations is the independent groups t-test appropriate (if inappropriate, why?):

a. The independent variable is infant birth weight at one week (normal vs high); the dependent variable is resting heart rate.

b. The independent variable is radiation treatment on throat cancer patients (after a low dose and then a high dose treatment); the dependent variable is white blood cell count.

c. The IV is infant birth weight (low vs normal vs high); the DV is number of days absent from school in first grade.

d. The IV is gender (male vs female); the DV is compliance vs noncompliance with a medication regimen.

e. The independent variable is married status (single vs married); the dependent variable is happiness measured on a scale from 1 to 50.

For which of the following situations is the dependent groups t-test appropriate (if not appropriate, why?)

a. The IV is presence or absence of conversation directed to comatose patients (same patients with and without conversation); the DV is the patients’ intracranial pressure.

b. The IV is birth type (home vs hospital); the DV is perceived functional ability of the patient 48 hours after surgery.

c. The IV is time since incarceration (1 month vs 3 months vs 6 months); the DV is body weight.

d. The IV is menopausal state (pre vs post) in the same women over time; the DV is attitudes toward menopause.

e. The IV is nap therapy for narcoleptics (same patients before vs after treatment); the DV is the type of nap they had the following week (had unplanned vs didn’t have unplanned nap).

Suppose we wanted to test the hypothesis that a control group of cancer patients (Group 1) would report higher mean pain ratings than an experimental group receiving special massage treatments (Group 2). Use the following information. Compute a t-statistic for independent groups: mean group 1 = 78.1 SD 42.1 n1 = 25, mean group 2 = 74.1 SD 39.7 n2 = 25. What are the degrees of freedom and the value of t? Using α=0.05 for a two-tailed test, is this t statistic significant? Show your calculations or StatCrunch output for full credit.

Write one or two sentences that could be used to report the results obtained for the t-test in Exercise 8.

For each of the following t values, indicate whether the t is statistically significant for a two-tailed test, at the specified alphas: a. t = 2.90, df = 25, α=0.01; b. t = 2.00, df = 25, α=0.05; c. t = 5.52, df = 10, α=0.01; d. t = 2.02, df = 20, α=0.

For each of the following situations, indicate whether ANOVA is appropriate; if not, why; and if yes, specify the type (one-way, repeated measures, etc.):

a. The IVs are ethnicity (Asian, White, African American, Hispanic) and gender (male vs female); the DV is serum cholesterol levels.

b. The IV is smoking status – smokers vs non-smokers; the DV is health-related hardiness as measured on a 20-item scale.

c. The IV is maternal breastfeeding status (daily, 1-3 times/week, doesn’t breastfeed); the DV is maternal bonding with infant.

d. The IV is treatment group for drug-induced shivering (extremity wraps vs high room temp vs normal room temp); the DV is myocardial oxygen consumption.

e. The IV is length of gestation (preterm, term, postterm) over time; the DV is blood pressure 10 minutes post delivery.

Suppose we want to compare the somatic complaints (measured on PSS) of three groups: non-smokers, smokers, and quitters. Using the data, conduct a one-way ANOVA to test if population means are equal. Using StatCrunch, determine group means, sums of squares, degrees of freedom, and F value. Is the F significant at alpha=0.05? Paste your work from StatCrunch.

Similarly, for the motorcycle MPG data of three models (X-1, B-1, Z), determine group means, sums of squares, degrees of freedom, and F. Is the F significant at alpha=0.05? Include your StatCrunch output.

Sample Paper For Above instruction

Probability calculations for rolling a four with two six-sided dice are fundamental in understanding outcomes in craps. Each die has six faces, numbered 1 through 6. The total number of possible outcomes when rolling two dice is 36, because each die roll is independent and has 6 outcomes, resulting in 6 x 6 = 36 combinations. To find the probability of rolling a sum of four, we identify all combinations that result in this sum: (1,3), (3,1), (2,2). There are three such outcomes. Therefore, the probability of rolling a four on a single roll is 3/36, which simplifies to 1/12 or approximately 0.0833. The probability of rolling a four three times in a row, assuming each roll is independent, is the cube of the single-roll probability: (1/12)^3 = 1/1728, approximately 0.000578. This means that the likelihood of this specific sequence occurring is very low, emphasizing the rarity of extraordinary streaks in dice games.

When comparing population means, the accuracy of sample estimates depends on the sample size and variability within the sample. Both Population A and Population B have the same mean height of 70 inches and a standard deviation of 6 inches. A larger sample size tends to produce a more precise estimate of the population mean because it reduces sampling variability. The sample from Population B, which has 50 individuals, is larger than the sample from Population A with 30 individuals. Statistically, an increased sample size improves the estimate's accuracy; hence, the sample mean from Population B, being based on 50 observations, is likely to yield a more accurate estimate of its population mean than the smaller sample from Population A.

Confidence intervals provide a range within which the true population parameter is expected to fall, with a certain level of confidence. The data on vein size in 50 patients shows a mean of 8.7 mm with an SD of 2.1. Using a t-distribution table for a 95% confidence level and degrees of freedom (df) = 49 (since n-1), the critical t-value is approximately 2.009. The margin of error (ME) is calculated as t (SD/√n): ME = 2.009 (2.1/√50) ≈ 2.009 0.297 = 0.597. Therefore, the 95% confidence interval is 8.7 ± 0.597, or approximately (8.10 mm, 9.30 mm). For a 99% confidence interval, the critical t-value for df=49 is approximately 2.704. The margin of error then is 2.704 0.297 ≈ 0.804, giving an interval of 8.7 ± 0.804, or (7.896 mm, 9.504 mm). These intervals provide a range where the true mean vein size likely resides, with higher certainty at the 99% level.

The t-test conducted on data assessing the effect of a new drug on resting heart rate compares the sample mean to the known population mean (72 bpm). Suppose the sample has a mean HR of 68 bpm with an SD of 8 bpm and a sample size of 20. The t-statistic is calculated as (sample mean - population mean) / (SD/√n): t = (68 - 72) / (8/√20) = -4 / (8/4.472) ≈ -4 / 1.789 ≈ -2.236. Using df = 19 and alpha = 0.05, the critical t-value for a two-tailed test is approximately ±2.093. Since |t| = 2.236 > 2.093, the result is statistically significant, indicating the drug may have an effect on reducing resting HR. The 95% confidence interval for the mean difference is (mean difference) ± t (SD/√n): -4 ± 2.093 1.789 ≈ -4 ± 3.744, thus from approximately -7.744 to -0.256. This interval indicates the true mean reduction in HR with 95% confidence, suggesting the drug's potential effectiveness.

Results summary: The t-test shows a significant reduction in resting heart rate following drug administration (t = -2.236, p

The independent groups t-test is appropriate in situations where two independent samples are compared. For example, in scenario (b), comparing white blood cell counts between patients receiving low and high radiation doses, the groups are independent, and the test is suitable. In contrast, scenario (a) involves the same patients measured before and after conversation, implying a paired or dependent data structure, making a dependent groups test more appropriate. Scenario (c) involves three groups, which would require ANOVA, not an independent t-test, due to multiple groups. The same logic applies to scenarios (d) and (e), where the independence of samples determines the appropriateness of the test.

The dependent groups t-test (paired t-test) is suitable when the same subjects are measured under different conditions. For example, scenario (d), where menopausal status is evaluated in the same women pre- and post-menopause, calls for a paired t-test because the data are dependent. Conversely, scenario (a), involving different patients with and without conversation, involves independent samples; thus, an independent t-test is more appropriate. For scenario (b), if the same patients undergo different blood tests at different times, a paired t-test would be suitable; if different groups are involved, an independent test applies. Scenarios involving repeated measures on the same subjects over time or conditions are best analyzed with dependent t-tests.

The t-statistic for comparing pain ratings between control and massage groups is calculated by: t = (mean1 - mean2) / √[(SD1^2 / n1) + (SD2^2 / n2)]. Plugging in the values: t = (78.1 - 74.1) / √[(42.1^2 / 25) + (39.7^2 / 25)] = 4 / √[(1774.81 / 25) + (1576.09 / 25)] = 4 / √[(71.3924) + (63.0436)] = 4 / √134.436 ≈ 4 / 11.585 ≈ 0.345. With degrees of freedom approximately calculated via the Welch–Satterthwaite equation, the degrees of freedom are around 47. Since the t-value is very small (≈0.345), and the critical t-value at alpha=0.05 is about 2.013, the test indicates no significant difference. The results suggest no evidence that the massage reduces pain compared to control in this sample.

The F-statistics obtained from ANOVA tests compare variances across groups to assess whether group means differ significantly. For example, in the somatic complaints study, calculations using StatCrunch produce an F-value: if the F exceeds the critical value at alpha=0.05 with the appropriate degrees of freedom, the null hypothesis of equal means is rejected, indicating significant differences among the groups. Similarly, for the motorcycle MPG data, a significant F-value suggests at least one model's mean MPG differs from the others. These tests are vital because they allow researchers to analyze multiple group comparisons simultaneously, controlling the error rate when examining multiple comparisons.

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