7-2 Estimating a Population Mean 333 3. Evaluate the margin of error using E = ta>2 # s> 1n. 4. Using the value of the calculated margin of error E and the value of the sample mean x, substitute those values in one of the formats for the confidence interval: x - E 6 m 6 x + E or x { E or 1x - E, x + E2. 5. Round the resulting confidence interval limits as follows: With an original set of data values, round the confidence interval limits to one more decimal place than is used for the original set of data, but when using the summary statistics of n, x, and s, round the confidence interval limits to the same number of decimal places used for the sample mean. EXAMPLE 2 Confidence Interval Using Peanut Butter Cups Listed below are weights (grams) of randomly selected Reese’s Peanut Butter Cups Miniatures. They are from a package of 38 cups, and the package label states that the total weight is 12 oz, or 340.2 g. If the 38 cups have a total weight of 340.2 g, then the cups should have a mean weight of 340.2 g>38 = 8.953 g. a. Use the listed sample data to find the point estimate of the mean weight of a Reese’s Peanut Butter Cup Miniatures. b. Use the listed sample data to construct a 95% confidence interval estimate of the mean weight of Reese’s Peanut Butter Cup Miniatures. continued t 5 0 ta/2 5 2.571 0.025 0.025 FIGURE 7-5 Critical Value tA,2 EXAMPLE 1 Finding a Critical Value tA,2 Find the critical value ta>2 corresponding to a 95% confidence level, given that the sample has size n = 6. SOLUTION Because n = 6, the number of degrees of freedom is n - 1 = 5. The 95% confidence level corresponds to a = 0.05, so there is an area of 0.025 in each of the two tails of the t distribution, as shown in Figure 7-5. Using Technology Technology can be used to find that for 5 degrees of freedom and an area of 0.025 in each tail, the critical value is ta>2 = t0.025 = 2.571. Using Table A-3 To find the critical value using Table A-3, use the column with 0.05 for the “Area in Two Tails” (or use the same column with 0.025 for the “Area in One Tail”). The number of degrees of freedom is df = n - 1 = 5. We get ta>2 = t0.025 = 2.571. YOUR TURN. Find the critical value for Exercise 2 “Degrees of Freedom.” Estimating Sugar in Oranges In Florida, members of the citrus industry make extensive use of statistical methods. One particular application involves the way in which growers are paid for oranges used to make orange juice. An arriving truckload of oranges is first weighed at the receiving plant, and then a sample of about a dozen oranges is randomly selected. The sample is weighed and then squeezed, and the amount of sugar in the juice is measured. Based on the sample results, an estimate is made of the total amount of sugar in the entire truckload. Payment for the load of oranges is based on the estimate of the amount of sugar because sweeter oranges are more valuable than those less sweet, even though the amounts of juice may be the same.
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