Question

You read about a survey in a newspaper and find that \(70 \%\) of the 250 people sampled prefer Candidate A. You are surprised by this survey because you thought that more like \(50 \%\) of the population preferred this candidate. Based on this sample, is \(50 \%\) a possible population proportion? Compute the \(95 \%\) confidence interval to be sure.

Short answer

The 95% confidence interval does not include 50%, suggesting 50% may not be a valid population proportion.

Step-by-step solution

Understand the Problem

We need to determine if the sample statistic significantly differs from the hypothesized population proportion of 50%. Calculate the 95% confidence interval for the sample proportion and check if this interval includes 50%.

Identify the Sample Proportion

The sample proportion of people who prefer Candidate A is calculated using \( \hat{p} = \frac{x}{n} \), where \( x = 175 \) (because 70% of 250 equals 175) and \( n = 250 \). \[ \hat{p} = \frac{175}{250} = 0.7 \] Thus, the sample proportion \( \hat{p} \) is 0.7.

Calculate the Standard Error

The standard error (SE) of the sample proportion is calculated as \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). \[ SE = \sqrt{\frac{0.7(1-0.7)}{250}} = \sqrt{\frac{0.21}{250}} = \sqrt{0.00084} \approx 0.0289 \]

Determine the Z-score for 95% Confidence Level

A 95% confidence level corresponds to a Z-score of approximately 1.96 for a two-tailed test.

Compute the 95% Confidence Interval

The confidence interval can be computed using \( \hat{p} \pm Z \times SE \), where \( Z = 1.96 \). \[ 0.7 \pm 1.96 \times 0.0289 = 0.7 \pm 0.0566 \] Thus, the confidence interval is approximately \( (0.6434, 0.7566) \).

Analyze the Result

The 95% confidence interval \( (0.6434, 0.7566) \) does not include 0.5. Therefore, it is unlikely that the true population proportion is 50%.

Key concepts

Sample Proportion

In statistics, the sample proportion is a valuable measure that estimates a characteristic of the population based on a sample. It is denoted by \( \hat{p} \) and calculated using the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successful outcomes in the sample, and \( n \) is the total sample size.

• In the given example, 175 out of 250 people prefer Candidate A, resulting in a sample proportion of \( \hat{p} = 0.7 \).
• This proportion reflects the preference in the sample and serves as an estimate of the population proportion.

Understanding the sample proportion helps us gauge whether our expectations about the population match reality. It acts as a snapshot of the sample's view but may vary when different samples are taken. This highlights the role of randomness and sampling variability in statistics.

Standard Error

The standard error of a statistic provides an estimate of the variability of the sample proportion from sample to sample. It is crucial for constructing confidence intervals and performing hypothesis tests. The standard error of the sample proportion \( \hat{p} \) is given by the formula\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

• In our exercise, with \( \hat{p} = 0.7 \) and \( n = 250 \), the standard error is calculated as approximately 0.0289.
• This value tells us how much the sample proportion is expected to fluctuate around the true population proportion.

A smaller standard error implies that the sample proportion is a more reliable estimate of the population proportion, while a larger one indicates greater uncertainty. Thus, the standard error is pivotal in determining the precision of our estimates.

Z-score

A Z-score translates a statistic, like a sample proportion, into the standard normal distribution, enabling comparison to a population mean or hypothesis. For confidence intervals, it indicates how far away the statistic lies from the mean in terms of standard deviations.

• For a 95% confidence interval, we use a Z-score of 1.96, representing the cut-off points separating the middle 95% of a normal distribution.

This score is instrumental in determining the required range to capture the true population proportion with a specified level of confidence. By multiplying the standard error by the Z-score, we obtain the margin of error, which is then added and subtracted from the sample proportion to create the confidence interval. Understanding the Z-score's application in this context allows statisticians to make informed decisions regarding the reliability of their samples.

Population Proportion Hypothesis

Hypothesis testing involves making an initial assumption about a population parameter and using statistical evidence to test its validity. In the context of this exercise, the hypothesis might be that precisely 50% of the population prefers Candidate A, known as the population proportion hypothesis.

• The null hypothesis is often the statement you want to test, which is here assumed to be \( p = 0.5 \).
• The alternative hypothesis is that the true population proportion is different from this hypothesized value.

After calculating the confidence interval based on the sample, we see that the interval \((0.6434, 0.7566)\) does not include 0.5, implying the hypothesis that 50% of the population supports Candidate A is unlikely.By applying confidence intervals and hypothesis testing, statisticians can draw conclusions about population parameters and assess the validity of assumptions, thereby contributing to informed decision-making.