Mathematics Question
STAT 5173/4173 Test 2 Spring 2024 1. Name: This data was collected at the University of California at Berkeley concerning gender patterns in graduate admissions. (Bickel and O’Connell, 1975). The cross-tabulated data for acceptance (no or yes) vs. gender is given in the table below for five separate departments. Scale: gender 1 (male), gender 2 (female) accepted 0 (no), accepted 1 (yes) Department 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 a. Berkley Data Gender Accepted 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 Count 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 For department 2: Find the odds (Accepted | male). 8 17 207 353 391 202 205 120 244 131 279 138 299 94 138 53 317 24 351 22 b. For department 2: Find the odds (Accepted |female). c. For department 2: Find the OR(Accepted) and interpret its meaning. It will be easier to interpret if you put the higher odds on top. d. For department 2: Find a 95% confidence interval for the true OR(Accepted). e. From the confidence interval, are the results statistically significant? Yes or No Forest scientists are concerned with the decline in forest growth throughout the world. One aspect of this decline is the possible effect of emissions from coal-fired power plants. The scientists in particular are interested in the pH level of the soil and the resulting impact on tree growth retardation. The scientist study various forests that are likely to be exposed to these emissions. An index of growth retardation is constructed from various measurements taken on trees with high values indicating greater retardation in tree growth. 2. Soil pH Growth Ret. 3.30 19.78 3.40 21.59 3.60 22.45 3.80 17.78 3.90 17.95 4.10 17.45 a. (Let Y be Growth Ret and X be Soil pH) Estimate the simple linear regression model. b. Predict Growth Ret when Soil pH is 4. c. Identify 𝑆𝜀 , the sample standard deviation of the regression line. d. ̂1 (this is the slope in the model in part (a)) in context of the Interpret the value of 𝐵 problem. e. Find a 95% confidence interval for 𝐵1. f. Test the following hypothesis: Ho: 𝐵1 = −6 𝑣𝑒𝑟𝑠𝑢𝑠 Ha: 𝐵1 ≠ −6 g. Find the 95% confidence interval for the average Growth Ret when Soil pH is 4. Interpret the confidence interval. h. Test the assumption that the residual can be modeled by the normal distribution. Make sure you setup the null and alternate hypothesis, report a p-value, and draw conclusions. 3. De Anza College is interested in the relationship between anxiety level and the need to succeed in school. A random sample of 400 students took a test that measured anxiety level and need to succeed in school. The table shows the results. De Anza College wants to know if anxiety level and need to succeed in school are independent events. Need to Succeed in High Med-high Medium Med-low School Anxiety Anxiety Anxiety Anxiety High Need 35 42 53 15 Medium Need 40 48 63 25 Low Need 4 5 11 15 Column Total 79 95 127 55 Need to Succeed in School vs. Anxiety Level Low Anxiety 10 17 17 44 Row Total 155 193 52 400 a. If the two variables are independent, how many students do you expect to have a High Need to Succeed in School and a Medium Level of Anxiety? b. Test the hypothesis that the “Need to Succeed in School” is independent of “Anxiety Level”. Make sure you setup the null and alternate hypothesis, report the chi-square statistic and p-value and draw conclusions. c. If there is a statistical association between “Need to Succeed in School” and “Anxiety Level”, what pattern(s) do you see?
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