According to Thomson Financial, through January \(25,2006,\) the majority of companies reporting profits had beaten estimates (Business Week, February 6, 2006). A sample of 162 companies showed that 104 beat estimates, 29 matched estimates, and 29 fell short. a. What is the point estimate of the proportion that fell short of estimates? b. Determine the margin of error and provide a \(95 \%\) confidence interval for the proportion that beat estimates. c. How large a sample is needed if the desired margin of error is \(.05 ?\)

To calculate the point estimate for the proportion that fell short of estimates, divide the number of companies that fell short by the total number of companies in the sample. Point Estimate = (Number of companies that fell short) / (Total number of companies) Point Estimate = 29 / 162 Point Estimate ≈ 0.179 The point estimate for the proportion of companies that fell short of estimates is approximately 0.179.

To calculate the margin of error, we need to find the standard error for the proportion that beat estimates and then use the Z-score for a 95% confidence interval. First, we will find the proportion that beat estimates: Proportion (p) = (Number of companies that beat estimates) / (Total number of companies) p = 104 / 162 p ≈ 0.642 Now, we will calculate the standard error (SE): SE = \(\sqrt{\frac{p(1-p)}{n}}\) SE = \(\sqrt{\frac{0.642(1-0.642)}{162}}\) SE ≈ 0.038 For a 95% confidence interval, the Z-score is 1.96. Now we can calculate the margin of error (ME): ME = Z-score × SE ME = 1.96 × 0.038 ME ≈ 0.075 A 95% confidence interval for the proportion that beat estimates is: Lower limit = p - ME = 0.642 - 0.075 ≈ 0.567 Upper limit = p + ME = 0.642 + 0.075 ≈ 0.717 The 95% confidence interval for the proportion that beat estimates is approximately (0.567, 0.717).

To determine the sample size needed for a desired margin of error (ME) of 0.05, we will use the formula: n = \(\frac{Z^2 \times p \times (1-p)}{(ME)^2}\) Using the proportion (p) we calculated earlier and the Z-score for a 95% confidence interval (1.96), we can plug in the values in the formula: n = \(\frac{(1.96)^2 \times 0.642 \times (1-0.642)}{(0.05)^2}\) n ≈ 245.47 Since we cannot have a fraction of a company, we need to round up to the nearest whole number. Thus, a sample size of 246 companies is needed to achieve a margin of error of 0.05.