Question

A car manufacturer is interested in learning about the proportion of people purchasing one of its cars who plan to purchase another car of this brand in the future. A random sample of 400 of these people included 267 who said they would purchase this brand again. For each of the three statements below, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1 : The estimate \(\hat{p}=0.668\) will never differ from the value of the actual population proportion by more than \(0.0462 .\) Statement 2 : It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0235 . Statement 3: It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0462 .

Short answer

Statement 1 is incorrect, as the actual margin of error is larger than 0.0462. Statement 2 is also incorrect, as it is possible that the difference between \(\hat{p}=0.668\) and the actual population proportion can be larger than 0.0235. Statement 3 is correct, as the likelihood of the difference between \(\hat{p}=0.668\) and the actual population proportion being larger than 0.0462 is quite small.

Step-by-step solution

Analyzing Statement 1

First, we need to calculate the margin of error, let's use the formula \(E = Z * \sqrt{\frac{{\hat{p}*(1 - \hat{p})}}{{n}}}\), where Z is the z-score which usually is 1.96 for a confidence level of 95%, \(\hat{p}\) is the sample proportion and n is the total sample size. Substituting the given values, we get \(E = 1.96 * \sqrt{\frac{{0.668*(1 - 0.668)}}{{400}}}\) which results in E = 0.0471. Comparing this with the value given in statement 1, 0.0462, it is clear that \(E > 0.0462\). Therefore, the statement 1 is incorrect because the margin of error is actually greater than \(0.0462\).

Analyzing Statement 2

The margin of error E that we calculated in Step 1 is greater than the difference value in Statement 2, i.e., 0.0235. Hence, it is noticeable that the difference between \(\hat{p}\) and the actual population proportion could be higher than 0.0235. Therefore, statement 2 is incorrect.

Analyzing Statement 3

In this case, the difference value in the statement, 0.0462, is less than the margin of error E that we calculated in Step 1. Therefore, it is unlikely that the difference between \(\hat{p}\) and the actual population proportion would be more than the calculated E (0.0471). Hence, Statement 3 is correct.

Key concepts

Population Proportion

The population proportion is a measure that tells us what fraction of a total population possesses a certain characteristic. In the context of the given exercise, it refers to the proportion of all customers of a car manufacturer who plan to purchase the same brand again in the future. The population proportion is often unknown and must be estimated using a sample.
It's important to understand that any sample only provides a snapshot of the whole, and the sample proportion, which represents the same characteristic within a group taken from the population, is used as an estimate for the population proportion.

Confidence Interval

A confidence interval provides a range of values that likely contain the true population parameter, such as the population proportion. It is constructed around a sample statistic and gives us an estimate of the degree of certainty—or confidence—we have in the accuracy of that sample statistic.
For example, when we say we are 95% confident, we are stating that if we were to take 100 different samples and compute a confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the actual population proportion. The width of the confidence interval relates to the certainty we have regarding the estimate. Narrower intervals suggest a higher precision, whereas wider intervals indicate less precision.

Z-Score

The z-score is a statistical measure that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If we know the standard deviation and the sample mean, we can calculate the z-score of a value to find how it compares to the population as a whole.
In the context of constructing confidence intervals and calculating margins of error, z-scores are used to determine how many standard deviations an estimate is away from the mean to ensure a certain level of confidence. The 1.96 z-score used in the exercise corresponds to a 95% confidence level, which is commonly used because it offers a good balance between certainty and precision.

Sample Proportion

The sample proportion, denoted as \(\hat{p}\), is the ratio of the number of favorable outcomes to the total number of cases in a sample. It is an estimate of the population proportion \(p\). For instance, in the car manufacturer exercise, the sample proportion \(\hat{p} = \frac{267}{400} = 0.668\) estimates the actual population proportion of people who would repurchase a car of the same brand.
While \(\hat{p}\) can give us a good estimate, it's vital to remember it's not the exact population proportion, just our best guess based on the sample data. This is why we have a margin of error to communicate the estimate's potential inaccuracy.

Margin of Error Calculation

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It gives us a range in which we think our estimate from the sample falls within relative to the true population value. Calculating the margin of error (\(E\)) incorporates the z-score for the desired confidence level, the sample proportion (\(\hat{p}\)), and the sample size (\(n\)).
The formula as shown in the exercise is \[E = Z * \sqrt{\frac{{\hat{p}*(1 - \hat{p})}}{{n}}}\].
After the calculation, the resulting margin of error can be applied to the sample proportion, yielding a confidence interval that we use to infer true population values. Understanding the proper application of E is crucial, as it is commonly misunderstood, which can lead to incorrect conclusions, as outlined in the exercise solutions.