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Section 9.1 A confidence Interval for a Mean DISCUSSION Statistics Our last lesson involved finding a confidence interval for a proportion. We found that we can substitute the proportion for a sample, _______, for the population proportion, _______, and obtained the relationship: In this lesson we are investigating the confidence interval for a ____________. Remember that the mean for the population is represented as ________. The mean for a sample is represented as ______. The standard error of the sampling distribution of a sample mean is ______ ______ , where ______ is ______ the sample size and ________ is the standard deviation of the population. If the sample size is large enough or the population is _________________ distributed the middle 95% would be represented as: As in our analysis with a confidence interval for a sample proportion, we often have only 1 sample to work with, and we do not know what the standard deviation is for the population. What we can do is use the standard deviation of the sample, _______. There is one problem with this, however. The sampling distribution of s is ________________ right. Although the average value of ______ is approximately equal to ________, because it is skewed right the ______________ interval will be too narrow, so the capture rate will be less than ________ if we use z = __________. To compensate for this, we increase the width of the ________________ interval by using a _______ value rather than _______. ______ is called a ___________ t value. It was developed by ______________________ in 1915 when he worked for the ___________________ Brewery. The company did not allow him to publish under his own name. In his articles, then, he used the pen name _________________. As a result, the _______ values are often called the ________________________________. The value to use for t* can be found from a table. In our book, the table is on page 826. Unlike z-scores, which give probabilities less than a certain value, the t-distribution gives values of the tail above the value. At the top of the table, we have the ___________ probability, ______. The bottom gives the confidence level between the upper and lower tail. For example, the 95% confidence level, C, corresponds with the tail probability, p, of ________ (2.5% upper and 2.5% lower). There are a number of values down the column for each tail probability (confidence level). The value used is determined by the sample size, _____. On the left side of the table, each row has a _____ value, which stands for _____________________________. The degrees of freedom is always 1 less than the __________________, _____. For example, if our sample size is 21 and we want to determine what t* value to use for a 95% confidence interval, we would go across the row where df is equal to ___________ = ________. The value we would use is _______________. Notice that as the sample size increases, the value of t* becomes closer and closer to the z value, for a confidence interval of 95% being _________. The t* value at a sample size of ________ is equal to the z value. The book example deals with comparing body temperatures for 10 males and 10 females. The goal is to determine reasonably likely temperature values for males and females. The data is given in the table on the next page: Body Temperature (0F) Male Female 96.8 97.8 97.4 98.0 97.5 98.2 97.8 98.2 97.8 98.2 97.9 98.6 98.0 98.8 98.1 98.8 98.6 99.2 98.8 99.4 We are told that the distributions for each are approximately normally distributed. The mean for the sample of 10 males is 97.88 and the sample standard deviation, s, is 0.555. We want to determine a 95% confidence interval for the males. You are 95% confident that the mean body temperature of men in this population is in the interval ___________________. For the females, the sample mean is 98.52 and the sample standard deviation, s, is 0.527. We want to determine a 95% confidence interval for the females. You are 95% confident that the mean body temperature of women in this population is in the interval ___________________. We might be tempted to say that the average body temperature of females is ___________ than that of males. We need to be careful, though, because the range of likely values has some _____________, so this might be an incorrect assumption. In drawing a conclusion, it is good practice to say something like “I am 95% confident that the mean of whatever you are studying is in the interval from lower value to higher value. The Meaning of “95% Confident” Saying you are 95% confident means that if you could take random samples repeatedly from the population and compute a confidence interval from each sample, in the long run 95% of these different ____________ would __________________ or ________________ _______. Margin of Error The margin of error for a confidence interval for a mean is Like that of the proportion, it is half the width of the _______________ interval. From the relation, you can see that as the sample size ______ increases, the margin of error _________________. Expressing the margin of error for our male and female temperatures, we would say that for men: ______________________ and for women: _______________________