Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Confidence Intervals I WEEK SIX This worksheet relates to chapter seven of the text book (Statistics for Managers 4th Edition). Please make sure you understand all of the questions we go through. Do not just copy down the answers and say that you will work it out later… it often doesn’t happen! DISCUSSION QUESTIONS 1. Why are interval estimates needed in addition to point estimates? 2. What are the differences between the t distribution and the Z distribution? When do we use them? 42 CALCULATION QUESTIONS 1. On the west bank of Cabbage Tree Creek live a very rare, very endangered species known as the woongoolamaloos. The harsh rains of December attracted fierce meerkats who ate many of the woongoolamaloos. So as to escape the wrath of the angry meerkats who suffered indigestion after their woongoolamaloo feast, the woongoolamaloos decide to move to a new shrub on the east bank of the river. Historical data suggests that the time it takes this species to cross the creek can be modeled as a normal distribution with a variance of 25 minutes. 35 woongoolamaloos were studied and the average time to travel across the creek was found to be 55 minutes. Find a 90% confidence interval for the true mean time it takes the woongoolamaloo population to cross the river and find safety and security under the new shrub. Interpret your confidence interval. 43 2. Approximately 200 people were abducted by aliens from Noosa beach on the eve of the 2006 New Year. Tragically, their brains were tampered with causing them to go progressively insane. Once the aliens had completed their tests, the 200 people were brought back to earth. A random sample of 90 of the 200 were analysed to work out how many days it took for the aliens’ surgery to take effect and cause the abductees to truly believe they were chickens (or other such farm animals). The analysis showed a mean of 10 days with a standard deviation of 2.1 days. Find a 95% confidence interval for the mean time period (in days) it took for the alien’s brain tampering to take effect. Interpret. 44 3. A survey of 2000 men in Queen Street Mall showed that 47% preferred blondes while 53% preferred brunettes (as for red heads… well they just couldn’t gain any support). Find a 92% confidence interval for the proportion of men who prefer blondes. Interpret the confidence interval. MULTIPLE CHOICE PRACTICE 1. A confidence interval shows (a) (b) (c) (d) a point estimate of the population mean an interval estimate of where the sample mean will lie the range which contains the population mean an estimate of the range in which the population mean will lie 45 2. The precision of confidence intervals is related to (a) (b) (c) (d) 3. $2567 $383 $6589 $2496 Final, June 2005 A market research company conducted a telephone survey of 765 households on behalf of a chain of new car dealers to determine the proportion of households seeking to purchase a new car in the next two years. An interval estimate of 0.076 and 0.112 was found. Determine the level of confidence that can be attached to this interval. (a) (b) (c) (d) 5. Final, June 2005 A statistical report states that the 95% confidence interval for the mean salary of new recruits in a company is between $25 500 and $27 000 obtained from a sample of 45. Calculate the population standard deviation. (a) (b) (c) (d) 4. the sample size and the confidence level the confidence level only the mean of the population and the confidence level the mean of the sample and the sample size 8.7% 95.6% 91.3% insufficient information to determine Final, June 2004 When comparing the t distribution (eg. For n = 20) with the Z distribution it is TRUE to say (a) (b) (c) (d) they are both normal curves the variances for both distributions are the same they have the same mean but the t distribution has more probability in the tails the t distribution has a smaller standard deviation than Z Final, June 2003 46 6. A random sample of customers buying petrol was selected. From this sample, the 95% confidence interval estimate for the mean amount of petrol purchased per customer for the city was calculated to be between 55 and 78 litres. This means that: (a) (b) (c) (d) 95% of all customers bought petrol in the given range. There is a 95% probability that the sample mean lies in this range. If another sample was taken there is a 95% chance that the confidence interval for the mean would be in the given range. There is a 95% probability that the mean amount of petrol purchased per customer for the city lies in the given range. Final, June 1998 7. Consider the following statements concerning confidence intervals: A. A disadvantage of point estimates, when compared to interval estimates, is that the increased precision achieved through taking a larger sample is not reflected in these estimates B. For a given sample size and population standard deviation, an increase in the confidence level will make the interval narrower. C. The only requirement for being able to use the standard normal variable to obtain the critical value for a confidence interval estimate for the population proportion is that the sample size be large (a) (b) (c) (d) (e) only A is true only C is true only A and B are true only B and C are true none of A, B and C are true 47 8. Consider the following of the Student’s t distribution: A. the variance of the t distribution decreases as the degrees of freedom decrease. B. The t statistic follows the t distribution, with the degrees of freedom being the sample size less one, provided that the random variable which is used to construct the statistic follows the normal distribution C. In practice, the t distribution can be used to determine the critical value for a confidence interval estimate for the population mean when the population variance is unknown, provided the sample size is sufficiently large and the distribution of the underlying population is not highly skewed (a) (b) (c) (d) (e) only B is true only C is true only A and B are true only B and C are true A, B and C are true 48 notes 49