Stat 226 Homework 5 Spring 2010 Due Thursday April 1 1. Normal Calculations. Creating condence intervals for the population mean combines many of the...
sampling sizes. an important, but offen underused, part of statistics uses our knowledge about how confidence intervals are created to estimate teh sample size needed to achieve a certain level of accuracy in our results. by making sure that a large enough sample will be collected before going through the process we can save time and money, with the added benefit of making you look smart in front of your boss! consider a population having standard deviation equal to 10. we wish to estimate the mean of this population.
a.) how large a random sample is needed to construct a 95% confidence interval for themean of this population with a margin of error equal to 1
b.) suppose that we now take a random sample of the size we have determined in part (a). if we obtain a sample mean x equal to 295, calculate the 95% confidence interval for the population mean.
c.) if i only required the interval in (a) to have a margin of error equal to 2, how large a random sample is needed (use the same confidence level)?
d.) suppose that we now take a random sample of the size we have determined in part c) if we obtain a sample mean x equal to 295, calculate the 95% confidence interval for the population mean. what is the interval's margin of error?
Stat 226Homework 5Spring 2010Due Thursday April 11.Normal Calculations. Creating conﬁdence intervals for the population mean combines manyof the subjects we covered in Chapter 1. These include: calculating the sample mean andcalculations with the Normal distribution. Calculating the sample mean is straight forward(especially with a calculator) so let’s remind ourselves of how both Normal calculations andbackward Normal calculations work by doing a few problems. Assume the distribution of IQscores in the adult population has a normal distribution with meanμ= 100 and standarddeviationσ= 15.(a) What percent of adults have an IQ below 75?(b) What percent of adults have an IQ between 82 and 118?(c) 5% of adults have an IQ higher than what value?(d) The central 95% of all adults have an IQ score between what two numbers?2.Critical Values. Now that we are comfortable again using backward Normal calculations, weare well on our way to being able to ﬁnd the critical values needed to create conﬁdence intervalsfor the population meanμ. As we learned in class, the critical valuez?in the formula for alevel C conﬁdence interval is the value that captures the central C% area under the standardNormal curve betweenz?andz?. Find the critical values for conﬁdence intervals with thefollowing conﬁdence levels:(a) 85%(b) 92%(c) 98%(d) What happens to the size of the critical valuez?as you increase the conﬁdence level?Intuitively, why does this make sense?3.Conﬁdence Intervals. Since we now know how to ﬁnd any critical value, we can combinethis knowledge with the results of a simple random sample to create conﬁdence intervals for thepopulation meanμ. Suppose for a random samplen= 25 measurements, we ﬁnd that ¯x= 50and we can assume that the population standard deviationσ= 10. Calculate conﬁdenceintervals forμwith the following conﬁdence levels:(a) 85%(b) 92%(c) 98%(d) What happens to the width of the interval as you increase the conﬁdence level? Intuitively,why does this make sense?1
4.How Long is The Wait?. A bank manager has developed a new system to reduce the timecustomers spend waiting to be served by tellers during peak business hours. The mean waitingtime during peak business hours under the current system is 10 minutes. The bank managerhopes that the new system will have a mean waiting time that is less than six minutes. Data collected during a trial run of the new system can be found in the JMP data ﬁleWaitTime.JMP.Assume that the population standard deviationσis known to be 2.5 minutes.(a) Use JMP to obtain an analysis of the distribution of wait times seen in this sample. Printand hand in this JMP output.(b) What is the sample mean wait time ¯x? What is the sample sizen?(c) Since we have a large sample, the shape of the distribution of wait times in this sampleshould be similar to the shape of the wait times in the population. Even if the populationdistribution is not Normal, what important statistical result tells us that the distributionof the sample mean is approximately Normal for large samples?(d) Calculate 95 percent and 99 percent conﬁdence intervals forμ.(e) Give the interpretation of the 95 and 99 percent conﬁdence intervals forμyou just calculated.(f) Using the 95% conﬁdence interval, can the bank manager be 95% conﬁdent thatμis lessthan six minutes? Explain.(g) Using the 99% conﬁdence interval, can the bank manager be 99% conﬁdent thatμis lessthan six minutes? Explain.(h) Based on your answers to parts f and g above, how convinced are you that the new meanwaiting time is less than the old mean time of 10 minutes? How convinced are you thatthe new mean waiting time is less than the desired time of six minutes?5.What is Conﬁdence?. In this problem, we are going to explore the idea of conﬁdence intervalsby looking at the meaning of the word conﬁdence. Trash bags have diﬀerent breaking strengths(in pounds) according to how much weight the bag will hold, on average, before breaking. Fora particular brand of trash bag, the breaking strength of the bags has a normal distributionwith meanμ= 50.5 pounds and a standard deviation ofσ= 1.65 pounds. What would happenif we took samples of sizen= 40 from this distribution? What would our conﬁdence intervalslook like? How many of our conﬁdence intervals would contain the true meanμ= 50.5 pounds?To explore these questions, open theCImean.JSLﬁle found in the Homework 5 folder inWebCT. This is a JMP script ﬁle and is similar to the module used in Homework 4 to explorethe sampling distribution of the sample mean statistic. On the left hand side of the screen,enter the information from our population above under the Population Characteristics heading.You can enter the Variable Name to beBreaking Strength. Under the Demo Characteristicsheading, enter a Sample Size of 40. Now click on the buttonDraw Samples.(a) On the upper right hand side of the screen, the histogram of the sample data is displayed,along with the summary statistics ¯xandsfor your sample. Write down the value of themean and standard deviation for your sample.2
Course Hero has all the homework and study help you need to succeed! We’ve got coursespecific notes, study guides, and practice tests along with expert tutors.

Educational Resources

Study Documents
Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.
Get oneonone homework help from our expert tutors—available online 24/7. Ask your own questions or browse existing Q&A threads. Satisfaction guaranteed!