Practice Test 2 (Ch 5 – 7) 1. Determine which of the following are valid probability distributions. If not, state why. If so, find the mean. a. X 0 1 2 3 4 P(X) 0.041 0.200 0.367 0.299 0.092 b. X 1 2 3 4 5 P(X) 0.001 0.020 0.105 0.233 0.242 c. X Email Face to face Text Chat P(X) 0.06 0.55 0.24 0.15 2. The table below describes the results of ten births from ten different sets of parents. Use this to find the probabilities. Number of girls x P(x) 0 0.001 1 0.010 2 0.044 3 0.117 4 0.205 5 0.246 6 0.205 7 0.117 8 0.004 9 0.010 10 0.001 Find the probability that: a. Getting exactly 8 girls in 10 births b. Getting at least 8 girls in 10 births. c. Is this an example of a discrete distribution? Why or why not. 3. Shawn and Maddie buy a property for $50,000 and spend $27,000 in repairs. They believe there is a 15% probability they can sell it for $120,000, 45% probability of selling it for $100,000, a 25% probability of selling it for $80,000, and a 15% probability of selling it for $60,000. Compute and interpret the expected value. 4. In a quick pick lottery game you spend $1 to pick a sequence of four digits (0 through 9), with repetition allowed. If the numbers drawn match the numbers on your ticket, you win $5,000. Compute and interpret the expected value. 5. A poll found that 42% of Americans believe in ghosts. A sample of 8 people is chosen at random, find the following probabilities: a. Exactly 3 people say they believe in ghosts. b. At least 6 people say they believe in ghosts c. Less than 7 people say they believe in ghosts. Practice Test 2 (Ch 5 – 7) 6. The probability of guessing the correct answer to a multiple choice question on the ACT is 20%. If a student guesses the answers on 7 problems on an ACT, find the following probabilities: a. The number of correct guesses is at least 5 b. The number of correct guesses is fewer than 2 c. The number of correct guesses is 7 7. In a poll of 5,000 people in a town, 78% said they are connected to the city water supply. Find the mean and standard deviation and give the range of usual values, μ – 2 σ to μ + 2 σ. 8. The reaction time, in minutes, of a certain chemical process follows a uniform probability distribution (rectangular), with 8 ≤ X ≤ 22. a. Sketch the rectangular uniform probability distribution and find P(x) b. What is the probability that the reaction time is between 12 and 15 minutes? 9. The ages of three patients are {58, 66, 84}. Consider all sample of two patients that can be chosen with replacement. Construct a table showing the distribution of sample means and their probabilities. 10. The average number of calories in a 1.5-ounce chocolate bar is 225 with a standard deviation of 10 calories. Find the probability: a. That a randomly selected 1.5-ounce chocolate bar will have more than 230 calories. b. That in a sample of 49 chocolate bars the sample average is between 200 and 220 calories. 11. The mean size of a widget is 20 inches with a standard deviation of 3 inches. Find the probability: a. A randomly selected widget will be smaller than 19 inches b. In a sample of 64 widgets, the sample average is between 17 inches and 19 inches. 12. A test is given to a sixth-grade class with an average score of 75 and a standard deviation of 8. If the top 10% of the scores are eligible for a $500 scholarship, what is the minimum score needed to be eligible for the $500 scholarship? 13. The average number of peanuts in a candy bar is 18 with a standard deviation of 5. If the number of peanuts is approximately normal, how many peanuts are in the middle 50% of bags? 14. According to a survey, 78% of Americans own a credit card. A random sample of 36 people is asked if they own a credit card. a. Show that the approximation is valid by verifying np ≥ 5 and nq ≥ 5. b. Find the probability that more than 25 people own a credit card. c. Find the probability that exactly 20 people own a credit card. Practice Test 2 (Ch 5 – 7) 15. The percentage of 18-24-year old eligible voters that voted in a recent primary was 15%. A survey of 100 people aged 18-24 was conducted. Note np ≥ 5 and nq ≥ 5 so a normal approximation is valid. Find the following probability: a. Use the binomial probability distribution to find the actual probability that exactly 25 of those surveyed did vote. b. Use the normal probability approximation to approximate the probability that exactly 25 of those surveyed did vote. c. Use the normal probability approximation to approximate the probability that more than 20 of those surveyed did vote. 16. Suppose a sample of n = 25 is taken and ๐ฅฬ = 5.9 ๐๐. It is known that ๐ = 1.1 ๐๐. a. What else must be true for you to be able to find a 95% confidence interval of the population mean? Why is this so? b. Would this require a Z-Interval or a T-Interval? Why? c. What is the best point estimate of ๐? d. Find the critical value ๐๐ e. Find the margin of error ๐ธ = ๐๐ f. ๐ √๐ Determine the 95% confidence interval. 17. Use the confidence interval (2.06, 5.22) to find the sample average and the margin of error. 18. A survey wishes to determine the 90% confidence interval for the mean age of an employee to within 1.5 years. The population of the age of employees is normally distributed with population standard deviation of 3 years. Find the minimum sample size needed to construct the interval estimate. Note ๐ = ( ๐๐ ๐ 2 ๐ธ ) 19. Suppose a sample of n = 36 is taken and ๐ฬ = 0.61. a. Verify that it is valid to construct a 99% confidence interval on the data. b. What is the best point estimate for the population proportion ๐ c. Find the critical value ๐๐ d. Find the margin of error ๐ธ = ๐๐ √ ๐ฬ ๐ฬ ๐ e. Determine the 99% confidence interval. 20. A survey wishes to construct a 90% confidence interval of the proportion of the population that supports Bill 172 before the state senate. The survey must be accurate to within 2% of the population proportion. Find the minimum sample size needed if a previous estimate found that ๐ 58% supported the legislation. Note ๐ = ๐ฬ ๐ฬ ( ๐ ) ๐ธ 2 Practice Test 2 (Ch 5 – 7) Use your calculator to find the following confidence intervals 21. The following data gives the number of minutes to complete an obstacle course. The population is normally distributed. 29 34 20 36 19 28 31 24 27 25 Determine the 90 % confidence interval of the true mean of the number of employees per division. 22. A sample of 100 drive thru times at a certain fast-food restaurant found the percentage of drive thru orders completed in under 90 seconds is 42%. Construct a 99% confidence interval of the proportion of drive thru orders that are completed in under 90 seconds. 23. In a sample of 46 new hires, the administrative costs for hiring a new employee have a mean of $ 41.60 with a population standard deviation of $ 7.50. Determine the 95% confidence interval of the true mean of new employee administrative costs. 24. A sample of 25 menthol cigarettes found a sample standard deviation of ๐ = 0.24 ๐๐ of nicotine. The amount of nicotine in a cigarette is normally distributed. Find a 90% confidence level of the population standard deviation ๐ using the ๐2 -table at the end of the exam and the formula: (๐ − 1)๐ 2 (๐ − 1)๐ 2 2 < ๐ < ๐2๐ ๐2 ๐ฟ Problems will require hypotheses, a claim, and whether there is evidence to support the claim. 25. A sample of 45 earthworm lengths were measured (in inches) with an average of 1.1 inches and a population standard deviation of 0.7 inches. Use α = 0.05 to test the claim that the average length of an earthworm is 1 inch. a. Set up a null and alternate hypothesis and identify the claim. b. Determine if this is a left, right, or two-tailed test c. Find the critical value(s) bounding the rejection region ๐๐ d. Determine the sample test statistic ๐ง = ๐ฅฬ −๐ ( ๐ ) √๐ e. Determine if you will reject or not reject the null hypothesis. f. Determine if there is enough evidence to support the claim. Practice Test 2 (Ch 5 – 7) 26. In a poll of 1025 potential voters, 592 said they would vote for Sponge Bob for President. Using a significance level of 0.01, test the claim that more than 50% of voters would vote for Sponge Bob. a. Set up a null and alternate hypothesis and identify the claim. b. Determine if this is a left, right, or two-tailed test c. Find the critical value(s) bounding the rejection region ๐๐ d. Determine the sample test statistic ๐ง ๐ฬ −๐ = (√ ๐๐ ) ๐ e. Determine if you will reject or not reject the null hypothesis. f. Determine if there is enough evidence to support the claim. 27. A random sample of 16 males found the average number of spoken words in a day was 16,576 words with a standard deviation of 7871 words. Use a significance level of ๐ผ = 0.05 to test the claim that the population of all males has a standard deviation that is greater than 7460. Note that the population is normally distributed. a. Set up a null and alternate hypothesis and identify the claim. b. Determine if this is a left, right, or two-tailed test c. Use the ๐2 -table at the end of the test to find the critical value(s) bounding the rejection region ๐๐ฟ2 , ๐๐ 2 , ๐๐ ๐๐๐กโ d. Find the sample test statistic ๐2 = (๐−1)๐ 2 ๐2 e. Determine if you will reject or not reject the null hypothesis. f. Determine if there is enough evidence to support the claim. Use your calculator and the p-value method to test the following claims. Problems will require hypotheses, a claim, and whether there is evidence to support the claim. 28. In a sample of 40 men, the mean weight is found to be 192.9 lbs with a population standard deviation of 39.8 lbs. Use α = 0.03 to test the claim that the average weight of men is 200 lbs. 29. The following data is the number of micrograms of lead per gram of medicine. Use a significance level of 0.05 to test the claim that the mean lead content in medicine is less than 14 micrograms per gram of medicine. The number of micrograms of lead per gram of medicine is normally distributed. 3.0 6.5 6.0 5.5 20.5 7.5 12 20.5 11.5 17.5 30. In a poll of 500 people in a town, 75% liked the new library. Use a significance level of ๐ผ = 0.01 to test the claim that at most 70% of the town like the library. Practice Test 2 (Ch 5 – 7)