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# 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 226 Homework 5 Spring 2010 Due Thursday April 1 1. Normal Calculations . Creating conﬁdence intervals for the population mean combines many of the subjects we covered in Chapter 1. These include: calculating the sample mean and calculations 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 and backward Normal calculations work by doing a few problems. Assume the distribution of IQ scores in the adult population has a normal distribution with mean μ = 100 and standard deviation σ = 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, we are well on our way to being able to ﬁnd the critical values needed to create conﬁdence intervals for the population mean μ . As we learned in class, the critical value z ? in the formula for a level C conﬁdence interval is the value that captures the central C% area under the standard Normal curve between - z ? and z ? . Find the critical values for conﬁdence intervals with the following conﬁdence levels: (a) 85% (b) 92% (c) 98% (d) What happens to the size of the critical value z ? 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 combine this knowledge with the results of a simple random sample to create conﬁdence intervals for the population mean μ . Suppose for a random sample n = 25 measurements, we ﬁnd that ¯ x = 50 and we can assume that the population standard deviation σ = 10. Calculate conﬁdence intervals for μ with 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 time customers spend waiting to be served by tellers during peak business hours. The mean waiting time during peak business hours under the current system is 10 minutes. The bank manager hopes that the new system will have a mean waiting time that is less than six minutes. Data col- lected during a trial run of the new system can be found in the JMP data ﬁle WaitTime.JMP . Assume that the population standard deviation σ is known to be 2.5 minutes. (a) Use JMP to obtain an analysis of the distribution of wait times seen in this sample. Print and hand in this JMP output. (b) What is the sample mean wait time ¯ x ? What is the sample size n ? (c) Since we have a large sample, the shape of the distribution of wait times in this sample should be similar to the shape of the wait times in the population. Even if the population distribution is not Normal, what important statistical result tells us that the distribution of the sample mean is approximately Normal for large samples? (d) Calculate 95 percent and 99 percent conﬁdence intervals for μ . (e) Give the interpretation of the 95 and 99 percent conﬁdence intervals for μ you just calcu- lated. (f) Using the 95% conﬁdence interval, can the bank manager be 95% conﬁdent that μ is less than six minutes? Explain. (g) Using the 99% conﬁdence interval, can the bank manager be 99% conﬁdent that μ is less than six minutes? Explain. (h) Based on your answers to parts f and g above, how convinced are you that the new mean waiting time is less than the old mean time of 10 minutes? How convinced are you that the 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 intervals by 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. For a particular brand of trash bag, the breaking strength of the bags has a normal distribution with mean μ = 50 . 5 pounds and a standard deviation of σ = 1 . 65 pounds. What would happen if we took samples of size n = 40 from this distribution? What would our conﬁdence intervals look like? How many of our conﬁdence intervals would contain the true mean μ = 50 . 5 pounds? To explore these questions, open the CImean.JSL ﬁle found in the Homework 5 folder in WebCT. This is a JMP script ﬁle and is similar to the module used in Homework 4 to explore the 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 be Breaking Strength . Under the Demo Characteristics heading, enter a Sample Size of 40. Now click on the button Draw Samples . (a) On the upper right hand side of the screen, the histogram of the sample data is displayed, along with the summary statistics ¯ x and s for your sample. Write down the value of the mean and standard deviation for your sample. 2
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