Chapter 8: Problem 39

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 \(.05 ?\)

Short Answer

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a. The point estimate for the proportion of companies that fell short of estimates is approximately \(0.179\). b. The 95% confidence interval for the proportion of companies that beat estimates is approximately (0.5674, 0.7166), with a margin of error of approximately 0.0746. c. A sample size of 246 companies would be needed to achieve a margin of error of 0.05.

Step by step solution

a. Point estimate for the proportion that fell short of estimates

To find the point estimate for the proportion of companies that fell short of estimates, we need to divide the number of companies that fell short (29) by the total number of companies in the sample (162). So, the point estimate of the proportion of companies falling short of estimates is: \(p = \frac{29}{162} \approx 0.179\)

b. Margin of error and 95% confidence interval for the proportion that beat estimates

To calculate the margin of error and provide a 95% confidence interval, first we need to find the proportion of companies that beat estimates, which can be obtained by dividing the number of companies that beat estimates (104) by the total number of companies (162): \(p_{beat} = \frac{104}{162} = 0.64198\) Next, we compute the standard error of the proportion using the formula: \(SE = \sqrt{\frac{p(1-p)}{n}}\) Where p is the proportion of companies that beat estimates, and n is the total number of companies in the sample. \(SE = \sqrt{\frac{0.64198(1-0.64198)}{162}} \approx 0.0381\) The critical value for a 95% confidence interval can be found in a Z-table, which is approximately 1.96. The margin of error can be calculated using the formula: Margin of Error = \(Z_{\alpha/2} \times SE\) Margin of Error = \(1.96 \times 0.0381 \approx 0.0746\) Now, to calculate the confidence interval, we can apply the following: Lower Limit: \(p -\) Margin of Error = 0.64198 - 0.0746 ≈ 0.5674 Upper Limit: \(p +\) Margin of Error = 0.64198 + 0.0746 ≈ 0.7166 Hence, the 95% confidence interval for the proportion of companies that beat estimates is approximately (0.5674, 0.7166).

c. Sample size needed for the desired margin of error of 0.05

To determine the sample size needed to achieve a desired margin of error, we need to use the following formula: \(n = \frac{(Z_{\alpha/2})^2 \times p \times (1-p)}{(E)^2}\) Where n is the sample size, \(Z_{\alpha/2}\) is the critical value for a specific confidence level, p is the proportion, and E is the desired margin of error. Using the values for a 95% confidence interval (\(Z_{\alpha/2} \approx 1.96\)), the proportion of companies that beat estimates (0.64198), and the desired margin of error (0.05): n = \(\frac{(1.96)^2 \times 0.64198 \times (1-0.64198)}{(0.05)^2}\) n = \(245.086\) Since the sample size must be a whole number, we round up to 246. Therefore, a sample size of 246 companies would be needed to achieve a margin of error of 0.05.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Estimate

When working with statistics, particularly when analyzing data sampled from a larger population, the proportion estimate is a fundamental concept that helps us understand the likely percentage of the population that exhibits a particular trait or characteristic. Think of it as taking a snapshot of a crowd to guess how many people are wearing hats – you count the hats in your snapshot to estimate the overall hat-wearing crowd.

In our example, the proportion estimate is calculated by dividing the number of companies that fell short of profit estimates by the total number of companies sampled. Here, with 29 out of 162 companies falling short, the proportion estimate is approximately 0.179. This figure is an indicator of what we can generally expect in the whole population, reflecting a common occurrence during a specific time period – much like identifying trends in fashion by looking at a representative group of people.

Margin of Error

Have you ever tried to measure something but couldn't get the exact size, say a piece of cloth without a ruler? That tiny bit of uncertainty is somewhat similar to what we call the margin of error in statistics. When we estimate proportions, we're not perfectly accurate due to the sample's limits. The margin of error gives us a range that says 'the true value is probably here, give or take a little.'

It's determined using the standard deviation of our proportion estimate and a 'z-score', which is like a yardstick for measuring how extreme our sample's results are compared to the norm. With a 95% confidence, our z-score is about 1.96 – assuming we're pretty close to average. For the proportion of companies that beat estimates, we're 95% confident that the true proportion is within about 0.0746 of our estimate – making our estimates give or take about 7.5%.

Sample Size Calculation

Let's say you're throwing a party and want to ensure everyone gets a slice of cake. You'd need to know how many guests are coming to decide the size of the cake you need, right? Similarly, when conducting a survey or study, you need to determine the number of participants (sample size) to ensure that your findings are reliable. This is where sample size calculation comes into play.

It's not just a guess; there's math involved. You'll need the margin of error you can tolerate, the confidence level you wish to have (often 95%), and the estimated proportion from a test or previous study. With these numbers, you solve the formula to find your 'cake size' – I mean your sample size – so that your study's results are trustworthy enough to make sound decisions. In this problem, to get our desired margin of error as tight as 5%, we'd aim for a sample size of 246 companies – ensuring everyone gets a 'taste' of our findings with good confidence.