# Elementary Probability and Statistics – Test 4 Show Working…

Elementary Probability and Statistics – Test 4 Show Working A die was rolled 300 times. The following frequencies were recorded. Outcome 1 2 3 4 5 6 Frequency 62 45 63 32 47 51 Do these data indicate that the die is unfair ? Use a 1% level of significance. How do you calculate expected frequencies for cells in a contingency table in a test of independence ? How are the degrees of freedom calculated ? Samples of US Congress members were taken. One sample was Democratic and the other Republican. The number of dollars spent on federal projects in each congressperson’s home district was recorded. Dollars Spent Party. < 5 billion 5-10 billion > 10 billion Row Total Democratic 8 15 22 45 Republican 12 19 16 47 Column Total 20 34 38 92 Use a 5% level of significance to test whether congressional members of each political party spent designated amounts in the same proportions. Pearson’s correlation coefficient, r, takes values in what range ? If the x and y labels on the variables are switched when calculating r, is it true or false that the value of r will change ? Consider the following data on study time (x) and subsequent test score ( y). x 20 15 12 10 7 4 y 100 85 75 72 64 40 Calculate Pearson’s correlation coefficient for this data. ( Show working ) Find the equation of the simple linear regression line for the data in question 5 ( Show working ). Calculate and interpret the coefficient of determination for the data and regression from question 5. a) For the regression in question 5, find the predicted test score for a student who studies for 12 hours b) Is the calculation in a) an example of interpolation or extrapolation ? Why ? For the regression line in question 6, interpret the slope and y intercept of the line. 10. A comparison is made between two bus lines to determine if arrival times of their regular hours from Denver to Durango are off schedule by the same amount of time. For 51 randomly selected buses, bus line A was observed to be off schedule an average of 53 mins, with standard deviation 19 mins. For 60 randomly selected runs, bus line B was observed to be off schedule an average of 62 mins, with standard deviation 15 mins. Do the data indicate a significant difference in average off-schedule times? Use a 5% level of significance. 11. A sample of 25 workers with employer provided health insurance paid an average premium of $6600 with a standard deviation of $800. Construct a 95% confidence interval for the mean premium amount paid by all workers who have employer provided health insurance. 12. Assume that IQ scores are normally distributed with mean 100 and standard deviation 15 points. If 100 people are randomly chosen, what is the probability that the sample mean will be between 98 and 104 points ? 13. It is known that about 64% of people who are murdered knew the person who murdered them. . ( p = 0.64 ). 63 unsolved murder cases are examined. Use the normal approximation to the binomial distribution to find the probability that at least 35 of the victims knew their murderers. 14. A company allows free returns of it’s products within one week after purchase. An average of 2 out of 10 products sold are returned within one week. Using the Poisson distribution, find the probability that exactly 7 out of 40 products sold on a given day will be returned within one week. 15. In a village in Hawaii, about 80% of the residents are of Hawaiian ancestry. Let n be the number people you meet until you encounter the 1st person of Hawaiian ancestry in the village. Write a formula for the probability distribution of the random variable n. Compute the probability that n=4. 16. In January in Hawaii, 60% of the days have surf at least 6 feet. At the start of the month, you pick 7 days randomly for vacation. What is the probability that at least 5 of your 7 vacation days have surf at least 6 feet ? 17. Roll an unfair die. X 1 2 3 4 5 6 P(X) 0.2 0.1 0.25 0.05 0.1 ? What does P(6) need to be for this to be a valid probability distribution? What is the expected value when this die is rolled?