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Statistics Assignment: Hypothesis Testing, Normality, Boxplots and T-tests

1. At a manufacturing plant, a new process has been proposed but will not be implemented unless it can be shown that the population mean process time is less than 15 minutes. The management would like to determine if there is sufficient evidence that the mean process time is less than 15 minutes.

A. State the null and alternative hypotheses.

B. Based on a sample of 30 observations of this process, the value of the test statistic was found to be -1.81 and the two-sided p-value was found to be 0.08. Find the appropriate one-sided p-value. Hint: if struggling with this question, draw this picture and label what you can

2.A national agency sets recommended daily allowances for many supplements. In particular, the allowance for zinc for adult men is 15 mg/day. The agency would like to determine if the average intake of zinc for adult men is greater than 15 mg/day. Suppose from a previous study they estimate the standard deviation to be 2 mg/day and they conjecture that the true population mean is 16 mg/day. The investigators plan to use a one-sample t-test with α=0.05.

A. Find the power with n=20 for the scenario above.

B. If the standard deviation was larger (more than 2), would the power be higher or lower than that calculated in part A?

C. If the sample size was larger (more than 20), would the power be higher or lower than that calculated in part A?

D. If we used α=0.01 (instead of 0.05), would the power be higher or lower than that calculated in part A?

E. Using a conjectured mean of 17 mg/day (instead of 16), would the power be higher or lower than that calculated in part A?

F. Return to the original scenario and find the sample size required to achieve 90% power. Remember to “round” up to an integer value

3. Use the data from Problem 5.29 which deals with lead concentrations in estuarine creeks. (don’t forget about, quote = “’” for these data)

A. Construct a histogram, qqplot and run SW test of normality. What do you conclude about the normality of the data? Do the various plots and tests agree?

B. Give the sample mean and median for this data.

C. Use the sign test to test the null hypothesis that the median is equal to 30. Give the pvalue and make a conclusion.

D. Give a 95% confidence interval for the median. Note: For consistency, please report the “Upper Achieved CI”. E. Give a (standard) 95% confidence interval for the mean.

F. It should be clear from the diagnostics in part A that the assumption of normality is not met. Hence the CI from previous question is suspect. Give a 95% bootstrap studentized confidence interval for the mean. Hint: See the BootStrap R code Example, then look at “boot example 2”. Considering use a different value for set.seed.

G. Assuming that cumulative lead exposure is of interest, would the mean or the median be of more interest.

4. Use the data from problem 6.43 which concerns noise levels of Wide and Narrow bodied jets. Note: This problem deals with dataset format and missing data. (don’t forget about, quote = “’” for these data)

A. Construct the side-by-side boxplots and include them in your assignment.

B. Give the sample means and standard deviations for each body type. Note: when calculating the mean or sd for the narrow bodied jets (which include NA values), you will need to use the na.rm = TRUE option. For example: mean(Jets$NarrowBodied, na.rm = TRUE)

C. Assuming equal variances, give the 95% confidence interval for the difference between the means. Based on this interval, can we conclude that there is a difference between the population means? Explain.

D. Considering the summary statistics in part B, is the pooled variance t-test or WelchSatterthwaite t-test preferred here? Justify your response using the rule of thumb from the notes.

E. Regardless of your answer from part D, run the pooled (equal variance) t-test to test H0: μ1-μ2=0 versus a two-sided alternative. Give the p-value and conclusion.