Probability and Statistics: Mean, Standard Deviation, and Confidence Intervals - Prof. Bra, Exams of Statistics

Download Probability and Statistics: Mean, Standard Deviation, and Confidence Intervals - Prof. Bra and more Exams Statistics in PDF only on Docsity! 1. Consider the following probability distribution for the discrete random variable X. X P (X) 4 .20 5 .15 6 .36 7 .14 8 .15 2. Find the mean of the discrete random variable X. 3. Find the standard deviation of the discrete random variable X. 4. Suppose that it is known that 20 percent of all college students abstain from drinking. A random sample of 6 students is selected. (a) Find the probability that exactly 3 students in this sample abstain from drinking. (b) Find the probability that at least 4 students in this sample abstain from drinking. (c) Find the probability that none of the students selected abstain from drinking. 5. The quality control department at Target randomly selects from a ship- ment a box that contains 20 DVD players and tests for defective DVD players. The inspector will randomly select 5 DVD players from the box and reject the shipment if more than 1 of the tested DVD players is de- fective. Unknown the the inspector, the box he chose contains 4 defective DVD players. (a) What is the probability that this shipment will be accepted? (b) What is the probability that this shipment will be rejected? 6. Find the following probabilities for the standard normal distribution. (a) P (−1 < z < 1.15) (b) P (1.12 < z < 2.22) (c) P (z < 1.75) 7. Let x be a normally distributed continuous random variable such that µ = 14 and σ = 5. Find the following probabilities. (a) P (11 < x < 15) (b) P (16 < x < 20) (c) P (x > 13) 1 8. Suppose that it is known that 20 percent of all college students abstain from drinking. A random sample of 200 students is selected. (a) Find the probability that exactly 40 students in this sample abstain from drinking. (b) Find the probability that between 35 and 50 students in this sample abstain from drinking. 9. The mean wage for 1000 employees at a company is $15.50 and the stan- dard deviation is $2.50. Let x be the mean wage of a random sample of 50 employees. (a) Find the mean µx of the sampling distribution. (b) Find the standard deviation σx of the sampling distribution. (c) Find the probability that x will be between $14.00 and $16.00. 10. It is known that 35 percent of the population has a certain characteristic. A random sample of 50 members of the population is chosen. Let p̂ be the proportion of the sample which has the characteristic. (a) Find the mean µ p̂ of the sample proportion. (b) Find the standard deviation σ p̂ of the sample proportion. (c) Find the probability that p̂ will be between .30 and .40. 11. A random sample of 100 members of the population is taken and it is found that of the 100 members, the mean age is 34.5 and the standard deviation is 8.3. (a) What is the point estimate for the population mean? (b) Find a 90 percent confidence interval for the population mean. (c) Find a 99 percent confidence interval for the population mean. (d) Suppose you want the 99 percent confidence interval to be x ± 1.5. How large should you make your sample size in order to ensure this? 12. In a random sample of 15 giraffes it is found that the mean length of neck is 6.25 feet. (a) What is a point estimate for the mean length of the neck of a giraffe? (b) Find a 90 percent confidence interval for the mean length of a giraffe’s neck. (c) Find a 99 percent confidence interval for the mean length of a giraffe’s neck. 13. A random sample of 100 members of the population finds that 45 of sample contain a certain characteristic. 2