91Ó°ÊÓ

The article "Most Customers OK with New Bulbs" (USA TODAY, February 18,2011 ) describes a survey of 1016 randomly selected adult Americans. Each person in the sample was asked if they have replaced standard light bulbs in their home with the more energy efficient compact fluorescent (CFL) bulbs. Suppose you want to use the survey data to determine if there is evidence that more than \(70 \%\) of adult Americans have replaced standard bulbs with CFL bulbs. Let \(p\) denote the population proportion of all adult Americans who have replaced standard bulbs with CFL bulbs. a. Describe the shape, center, and variability of the sampling distribution of \(\hat{p}\) for random samples of size 1016 if the null hypothesis \(H_{0}: p=0.70\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.72\) for a sample of size 1016 if the null hypothesis \(H_{0}: p=0.70\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.75\) for a sample of size 1016 if the null hypothesis \(H_{0}: p=0.70\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.71 .\) Based on this sample proportion, is there convincing evidence that more than \(70 \%\) have replaced standard bulbs with CFL bulbs, or is this sample proportion consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.

Step by step solution

01

Shape, center, and variability of the sampling distribution

Assuming the null hypothesis \(H_{0}: p=0.70\) is true, the sampling distribution of \(\hat{p}\) should follow a normal distribution because we have a sufficiently large sample size (1016). The center of the distribution is the mean, which is equal to the population proportion of the null hypothesis, \(p=0.70\). The variability of the sampling distribution can be measured by the standard deviation, which can be calculated as: \[\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\] where \(n\) is the sample size and \(p\) is the proportion assumed in the null hypothesis. Substitute the values \(p=0.70\) and \(n=1016\): \[\sigma_{\hat{p}} = \sqrt{\frac{(0.70)(1-0.70)}{1016}} \approx 0.0146\] So, the sampling distribution of \(\hat{p}\) is approximately normal with a mean of \(0.70\) and a standard deviation of \(0.0146\).

02

Evaluate the sample proportion of 0.72

Now, let's compare the given sample proportion of \(\hat{p} = 0.72\) with the mean of the null hypothesis, \(p=0.70\). First, find the z-score: \[z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}\] Substitute the values: \[z = \frac{0.72 - 0.70}{0.0146} \approx 1.37\] Since the z-score is 1.37, which falls within the range of typical values (between -1.96 and 1.96) for a 95% confidence interval, we would not be surprised if the sample proportion is 0.72 given that the null hypothesis is true.

03

Evaluate the sample proportion of 0.75

Similarly, let's find the z-score for the sample proportion of \(\hat{p} = 0.75\): \[z = \frac{0.75 - 0.70}{0.0146} \approx 3.42\] The z-score 3.42 falls outside the range of typical values for a 95% confidence interval. We would be surprised to see a sample proportion of 0.75 if the null hypothesis were true. In this case, we might consider rejecting the null hypothesis.

04

Evaluate the observed sample proportion of 0.71

Finally, let's find the z-score for the actual observed sample proportion of \(\hat{p} = 0.71\): \[z = \frac{0.71 - 0.70}{0.0146} \approx 0.68\] The z-score of 0.68 falls within the range of typical values for a 95% confidence interval. Based on this sample proportion (0.71), the observed value is consistent with the null hypothesis and we don't have convincing evidence that more than \(70\%\) of the adult American population have replaced standard bulbs with CFL bulbs.

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Key Concepts

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

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a foundational concept in statistical hypothesis testing. It represents a statement or idea that there is no effect or no difference, and it serves as a starting point for statistical analysis. In the context of our exercise, the null hypothesis \(H_0: p=0.70\) suggests that the true proportion of adult Americans who have switched from standard light bulbs to CFL bulbs is exactly 70%.

The null hypothesis is crucial because it provides a benchmark against which we can compare our sample data. When analyzing data, we often aim to either reject or fail to reject the null hypothesis based on statistical evidence. This process is vital in determining whether the trends or differences observed in your data are statistically significant or whether they could have occurred due to random chance. By assuming \(H_0\) to be true, statisticians can apply appropriate statistical tests to ascertain the validity of their observed findings.

Population Proportion

Population proportion refers to the ratio or fraction of individuals in the entire population that possess a particular characteristic or trait. In this exercise, the population proportion \(p\) corresponds to the percentage of all adult Americans who have swapped out their standard light bulbs for CFL ones.

This parameter is important in statistics because it helps researchers understand the underlying preferences or behaviors within the whole population. However, as it might be impractical to collect data from every member of the population, researchers use sample proportions (represented as \(\hat{p}\)) to make inferences about the population proportion. The closer the sample proportion \(\hat{p}\) is to the population proportion \(p\), the more reliable the inferences are likely to be.

In hypothesis testing, the given value of the population proportion under the null hypothesis (\(p_0\)) serves as a reference value for evaluating the sample data. In our scenario, the null hypothesis proposes that the population proportion \(p\) is 0.70, meaning 70% of adult Americans use CFL bulbs.

Z-score

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean. A Z-score enables researchers to determine how unusual or typical a particular sample observation is relative to the assumed population mean.

In our example, when checking if \(\hat{p} = 0.72\) or \(\hat{p} = 0.75\) is significant, we used the formula:

\[ z = \frac{\hat{p} - p}{\sigma_{\hat{p}}} \]

This formula helps calculate the Z-score by subtracting the hypothesized population proportion \(p\) from the sample proportion \(\hat{p}\), then dividing the result by the standard deviation of the sample proportion \(\sigma_{\hat{p}}\).

By assessing the Z-score, we can determine how extreme the sample result is, which assists in making decisions about the null hypothesis. Typically, a Z-score within the range of around -1.96 to 1.96 (for a 95% confidence interval) suggests the sample result is not statistically significant, thus supporting the null hypothesis.

Normal Distribution

Normal distribution, also known as a Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable. It is one of the most vital concepts in statistics due to its unique properties and prevalence across various natural phenomena.

The classic bell-shaped curve illustrates normal distribution. It is symmetrically centered around the mean, which represents the peak of the curve. A fundamental property of normal distribution is that approximately 68% of data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

In our exercise, we assume that the sampling distribution of the sample proportion \(\hat{p}\) follows a normal distribution because of the large sample size (1016). When sample sizes are large, the distribution of sample means behaves in a way that closely approximates normal distribution, regardless of the shape of the population distribution. This property is known as the Central Limit Theorem. Here, the normal distribution provides a basis for conducting statistical tests, such as calculating Z-scores to evaluate the null hypothesis.