STAT1070 Statistics

Question 1.

The datafile cholesterol.sav provides cholesterol measurements (in mg/dL). They are from heart attack patients 4 days following the attack and for a control group of people who had not hada heart attack.

(a) Is there a difference in the average cholesterol levels between the control group and the group of heart attack patients? Be sure to state the null and alternative hypotheses, the observed test statistic, the null distribution, the p-value, decision and an appropriate conclusion in plain language.

(b) Construct a 98% confidence interval for the difference in average cholesterol level between heart attack patients 4 days after the attack and the control group. Write a sentence interpretting this interval.

(c) List the assumptions you made to perform the analysis in parts (a) and (b). Do the data satisfy these assumptions? Provide reasons why or why not for each assumption.

Question 2.

Archaeologists have been interested in determining whether the breadth of Egyptian skulls has changed gradually over time, because this might indicate that interbreeding occurred with immigrant populations. A random sample of 30 skulls was taken from 5 different eras (4000 B.C., 3300 B.C., 1850 B.C., 200 B.C. and 150 A.D.), and the maximal skull breadth was recorded for each (in millimetres). The data are available in the file skulls.sav.

The variables in the file are:

(a) Is there evidence that different eras have different average maximal skull breadths? Conduct the appropriate test in SPSS and include relevant output. Be sure to state the null and alternative hypotheses, the observed test statistic, the null distribution, the p-value, decision and an appropriate conclusion in plain language.

(b) If appropriate, identify which eras have significantly different average maximal skull breadths, with reference to included SPSS output. If post-hoc tests are not appropriate, explain the purpose of a post-hoc test and why it is not appropriate in this example.

(c) Report the 95% pairwise confidence interval for the mean difference in skull breadths for the 4000 B.C. and 150 A.D. eras. Interpret this interval with a sentence in plain language.

(d) What are the assumptions of your analysis in part (a)? Are these assumptions met? Justify why or why not for each assumption, with appropriate references to SPSS output where needed.

Question 3.

Urine specific gravity (USG) is a measure of the weight of solids in water. Urine osmolality is ameasure of the concentration of the urine. Osmolality and USG measurements are usually highly positively correlated, however, when large and heavy molecules (such as glucose and protein) are present in the urine the results can diverge. The USG test is easier and more convenient, and is part of a routine urinalysis.1 Osmolality is considered a more exact measurement of urine concentration than USG, however, USG is often used by clinicians in routine practice to predict urine osmolality.2 In the questions that follow, you will explore the idea of using USG to predict urine osmolarity in the urine.sav data on 79 urine specimens that you first encountered in Assignment 1.3 In the SPSS file the variable gravity represents specific gravity and osmo represents the osmolarity (mOsm).

(a) Generate an appropriate scatter plot with a fitted regression line.

(b) Is there a statistically significant positive linear relationship between specific gravity and osmolality? Be sure to state the null and alternative hypotheses, test statistic, null distribution, p-value, decision and an appropriate conclusion in plain language.

(c) State the assumptions necessary for a regression analysis to be appropriate. State whether each of them is satisfied with a brief justification.

(d) Write down the equation for the estimated regression line and provide an interpretation of the slope coefficient.

(e) Predict the osmolality for a USG value of 1.025.

(f) Write down the R2 value for this regression and give an interpretation.

(g)Based on the R2 value do you think that using USG to predict osmolarity is a sensible thing to do? Why or why not.