PHA 6935 Applied Statistics

1. A random sample of 5 year old children is evaluated for being left- or right-handed. The sample size is n = 100 children of which 79 are right-handed, 15 are left-handed, and 6 use both hands equally. According to genetic analyses, the expression of the dominant gene for handedness is highly correlated with actual handedness. Test the following hypothesis based on the sample given above at a significance level of α = 0.05. (5 marks)

H0: p1= p2= p3= 0.33;

H1: at least two of the probabilities are not the same

1. The distribution of grades for various classes taught by different professors is given in the following table. Test the null hypothesis that the grade distributions are homogenous for these three professors at a significance level of α = 0.05. (11 marks)
 Professor Grade Einstein Rutherford Bohr A 18 36 20 B 25 44 15 C 85 73 82 D 17 12 8
1. Define in your own words the following words: (4 marks)
1. Sensitivity
2. Specificity
3. Accuracy
4. Test of homogeneity
2. Determine the sensitivity, specificity, and accuracy for the following analytical test data that detected the presence of morphine using the Marquis test. (5 marks)
 Expected results positive condition negative condition Actual morphine presence positive outcome 15 2 negative outcome 4 23

Would you conclude that the test is able to detect both true positive and true negative samples? Provide support for your answer.

1. The number of violent crimes reported to police on random selected days for 3 different cities is shown in the following table. Formulate the null and alternative hypothesis and calculate the F-test statistic. Use a significance level of α = 0.05 to reach a conclusion. [9 points]
 City New Amsterdam Old Brussels Future Toronto 5 2 8 9 4 12 12 1 10 3 13 3 9 7 9 7 6 14 13
1. The following ANOVA table is provided. [4 points]
 Source of variation Degrees of freedom Sum of squares Between 4 200 Within 45 3547

Calculate the following values assuming that all groups have the same sample size:

1. MSB and MSW
2. Number of groups and number of samples in each group
3. F-test statistic
4. Fcrit for α = 0.05. What is the conclusion?
1. Four catalysts that may affect the concentration of one component in a three-component liquid mixture are being investigated. The following concentrations are obtained. [12 points]
 Catalyst 1 2 3 4 58.2 56.3 50.1 52.9 57.2 54.5 54.2 49.9 58.4 57.0 55.4 50.0 55.8 55.3 51.7 54.9

Determine if the 4 catalysts yield the same concentration of the component. If they do not, then use a post-hoc multiple comparison of your choice to determine which pairs of catalysts are different. Use a significance level of α = 0.05 for all your calculations.