set 4 part 2 (Correlation and Regression) Name -------------------------------- 1. Suppose the following table represents a SRS of hospitals enrolled in a study looking at the relationship between number of beds, and number of physicians working full time at the facility. Number of Beds (X) 589 517 354 512 366 597 644 Number of Physicians Employed Full Time (Y) 948 931 684 846 602 987 962 N N* Mean SE Mean StDev Number of Beds (X) 7 Number of Physicians Emp 7 Minimum 0 511.3 0 851.4 Q1 Median 42.8 113.2 57.0 150.9 Q3 354.0 602.0 366.0 684.0 517.0 931.0 597.0 962.0 A) Calculate basic descriptive statistics for both your X and Y variables. (Mean, standard deviations). B) Calculate a correlation coefficient describing the relationship between number of beds and number of full time physicians employed at a SRS of hospitals. Ans: A) MeanX=511.29, sdX=113.196, MeanY=851.43, sdY=150.853 B) r=0.945 Pg. 337 2. Suppose a researcher is interested in the relationship between BMI (kg/m2) and fasting glucose levels (mg/dL). The researcher is conducting a cross-sectional study with prevalent data in order to quantify the relationship in the greater population. In order to do so, the researcher plans to conduct a formal hypothesis test. The following table includes measurements for both BMI and fasting glucose in a SRS of individuals. Write out your null and alternative hypotheses. Assuming all necessary conditions are met, carry out a formal test to determine if there is a correlation between BMI and fasting glucose. Interpret your p-value. BMI (kg/m2) Fasting Glucose (mg/dL) 32.4 112 29.7 116 31.5 114 32.1 117 30.2 114 30.4 113 Ans: Ho: =0 Ha:0 t statistic =-0.315 pvalue=0.768 We fail to reject the Ho that there is no linear relationship between the two variables. Pg. 344 3. A researcher is interested in the relationship between years or education and annual income. Suppose the hypothesis is that the more education you receive, the higher your annual income will be. In order to test this hypothesis, the researcher generated a SRS of participants enrolled in the study. The information about the SRS is presented below. Years Education 16 16 21 16 12 12 18 Income 63,256 61,578 112,589 74,869 43,765 49,653 89,653 The following computer output summarizes the regression equation and relevant statistics: Coefficientsa Model Unstandardized Standardized Coefficients Coefficients B Std. Error Beta (Constant) -43711.113 15096.891 1 edu 7219.286 936.012 .960 a. Dependent Variable: income A) B) C) D) t Sig. Construct the regression equation for the relationship between the two variables. Write out your null and alternative hypothesis for the t-test of slope coefficient. Calculate your test statistic. Determine your p-value and interpret your results. Ans: A) annual income=-43711.113+7219.286(education) B) Ho: =0, Ha: 0 C) t=b/SEb, t=7.713 D) Pvalue=0.001, there is a significant association between years of education and annual income Pg. 357 4. Suppose a researcher is interested in the relationship between fasting glucose levels (mg/dL) and systolic blood pressure (mmHG). In order to test the hypothesis, the researcher conducts a SRS of individuals enrolled in his cross-sectional study. The following table summarizes the participants in the sample. Fasting Glucose Level (mg/dL) 119 123 131 124 109 119 117 Systolic Blood Pressure (mmHG) 126 135 142 138 115 121 124 The following is a computerized output representing detailing the regression equation for the relationship between the two variables. SystolicBP=-34.991+1.361(FastingGlucoseLevel) A) Calculate SY|X B) Construct a 95% Confidence interval for the b in the regression equation above. C) Does your interval present evidence to reject or retain the null hypothesis that there is no relationship between fasting glucose level and systolic blood pressure? Ans: A) 3.62 B) (0.802, 1.920) C) Yes, there is evidence to reject such a Ho, as 0 is not within the interval Pg. 356