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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 0.046 . Statement 2 : It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.024 . Statement 3: It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.046 .

Step by step solution

01

Calculate the estimated proportion \(\hat{p}\)

To calculate the estimated proportion, we divide the number of people who said they would purchase this brand again by the total number of people in the sample: \(\hat{p} = \frac{267}{400} = 0.668\)

02

Calculate the margin of error

We will use a 95% confidence interval, which corresponds to a z-score of 1.96. The margin of error (ME) is given by the following formula: \( ME = z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \) where \(z\) is the z-score, \(\hat{p}\) is the estimated proportion, and \(n\) is the sample size. In this case: \( ME = 1.96 \times \sqrt{\frac{0.668(1-0.668) }{400}} \) \( ME \approx 0.046 \) Now let's analyze each statement.

03

Statement 1: Analysis

The statement is stating that the estimate \(\hat{p}=0.668\) will never differ from the value of the actual population proportion by more than 0.046. This statement is incorrect as the margin of error only provides a 95% confidence interval, meaning that there is a 5% chance that the actual population proportion will be outside this range.

04

Statement 2: Analysis

The statement is stating that it is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.024. Since the margin of error is approximately 0.046, this statement is incorrect. It is unlikely (95% confidence) that the estimate differs from the population proportion by more than 0.046, not 0.024.

05

Statement 3: Analysis

The statement is stating that it is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.046. This statement is correct, as the margin of error we calculated is approximately 0.046, which corresponds to a 95% confidence interval. This means it is unlikely (with 95% confidence) that the estimate differs from the population proportion by more than 0.046.

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Key Concepts

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

Confidence Interval

A confidence interval provides a range of values that is used to estimate the true population parameter. In the context of our exercise, this interval is applied to the population proportion, which allows us to say with a certain level of confidence how close our sample proportion (\(\hat{p} = 0.668\) in our case) is to the actual population proportion. The confidence level (commonly set at 95% or 99%) expresses how confident one is in this range capturing the true population parameter. For a 95% confidence interval, we typically use the z-score of 1.96 in calculations. This indicates that 95% of the time, the interval we calculate should contain the true population proportion.

Margin of Error

The margin of error (ME) quantifies the amount of random sampling error in a survey's results. Specifically, it tells us how much the estimated proportion could differ from the actual population proportion. In our example, the margin of error is approximately 0.046. This means there's a 95% chance the true proportion is within 0.046 of the estimated proportion, \(\hat{p}\).
The margin of error is calculated using the formula:
\( ME = z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)
where \(z\) is the z-score, \(\hat{p}\) is the sample proportion, and \(n\) is the sample size. The larger the sample size, the smaller the margin of error, indicating a more precise estimate.

Population Proportion

Population proportion refers to the fraction of the entire population that possesses a certain characteristic. In this exercise, it pertains to the proportion of car buyers who would repurchase from the same brand. The estimated population proportion is \(\hat{p} = 0.668\), derived from the sample data of 267 out of 400 individuals expressing their intent to buy again.
This proportion gives us an initial insight into consumer loyalty without surveying the entire population.
It is crucial as it serves as the basis for further statistical inference, such as establishing the confidence interval and calculating the margin of error.

Random Sample

A random sample refers to a subset of individuals chosen from a larger population, where each member of the population has an equal chance of being selected. This randomness ensures the sample is representative, reducing bias.
In our case, a random sample of 400 car buyers was used to estimate the wider population's behavior or opinions.
A properly random sample is crucial for the reliability of statistical inference. If the sample isn't random, it may not accurately reflect the population's characteristics, which can skew results and lead to incorrect conclusions about the population proportion, confidence intervals, and the margin of error.