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A poll conducted in 2013 found that \(52 \%\) of U.S. adult Twitter users get at least some news on Twitter, and the standard error for this estimate was \(2.4 \%\). Identify each of the following statements as true or false. Provide an explanation to justify each of your answers. (a) The data provide statistically significant evidence that more than half of U.S. adult Twitter users get some news through Twitter. Use a significance level of \(\alpha=0.01\). (b) Since the standard error is \(2.4 \%\), we can conclude that \(97.6 \%\) of all U.S. adult Twitter users were included in the study. (c) If we want to reduce the standard error of the estimate, we should collect less data. (d) If we construct a \(90 \%\) confidence interval for the percentage of U.S. adults Twitter users who get some news through Twitter, this confidence interval will be wider than a corresponding \(99 \%\) confidence interval.

Step by step solution

01

Hypothesis for Part (a)

We start by defining the hypotheses. The null hypothesis (\(H_0\)) is that 50% or less of U.S. adult Twitter users get some news on Twitter, \(H_0:p \leq 0.5\). The alternative hypothesis (\(H_a\)) is that more than 50% of U.S. adult Twitter users get some news on Twitter, \(H_a:p > 0.5\).

02

Calculate the Test Statistic for Part (a)

The formula for the test statistic for a proportion is \(z = \frac{\hat{p} - p_0}{SE}\), where \(\hat{p} = 0.52\), \(p_0 = 0.5\), and \(SE = 0.024\). Plug these values into the formula: \[z = \frac{0.52 - 0.5}{0.024} = \frac{0.02}{0.024} \approx 0.8333\]

03

Determine Significance for Part (a)

Using a significance level of \(\alpha = 0.01\), we compare the test statistic with the critical value from the standard normal distribution. For \(\alpha = 0.01\), the critical value is approximately 2.33. Since \(0.8333 < 2.33\), we do not reject the null hypothesis at \(\alpha = 0.01\). Thus, the statement is false.

04

Analyze Part (b)

The standard error of \(2.4\%\) does not relate to the percentage of U.S. adult Twitter users included in the study. It estimates the standard deviation of the sample proportion. Therefore, the statement claiming that \(97.6\%\) were included in the study is false.

05

Analyze Part (c)

To reduce the standard error of the estimate, we should collect more data, not less. This is because the standard error (\(SE\)) is inversely proportional to the square root of the sample size (\(n\)). Therefore, the statement is false.

06

Analyze Part (d)

A \(90\%\) confidence interval is narrower than a \(99\%\) confidence interval. Higher confidence levels require wider intervals to ensure that they capture the true parameter with higher probability. Therefore, the statement is false.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a method used in statistics to make inferences about a population parameter based on a sample statistic. It involves testing an assumption (the hypothesis) about a parameter. The process begins by forming two hypotheses:

• The null hypothesis (\(H_0\)
) that represents the status quo or a statement of no effect. In this exercise, it states that 50% or fewer U.S. adult Twitter users get news from Twitter.
• The alternative hypothesis (\(H_a\) that indicates a new effect or difference, in this case, more than 50% of U.S. adult Twitter users obtain news from Twitter.

The test statistic is calculated to make decisions between these two competing hypotheses. For our example, since the calculated test statistic is approximately 0.8333 and the critical value at a 1% significance level is around 2.33, the null hypothesis isn't rejected. Hence, there's not enough statistical evidence to support the claim that more than half of the users get their news from Twitter at this significance level.
Hypothesis testing helps determine if observed results are statistically significant or due to random chance.

Confidence Intervals

Confidence intervals provide a range of values for a population parameter, such as a population proportion, estimated from a sample statistic. This range is constructed so that it contains the true parameter with a specified confidence level, like 90% or 99%.
To construct a confidence interval, you can use the formula:\[\hat{p} \pm z^* \cdot SE\]where \(\hat{p}\) is the sample proportion, \(z^*\) is the z-value corresponding to the desired confidence level, and \(SE\) is the standard error.A higher confidence level leads to a wider confidence interval because we require a broader range to ensure the true population parameter is captured within it. This reminds us that certainty comes with a trade-off: the more confident we want to be, the wider our interval must be.

Standard Error

The standard error (SE) is a statistical measure that reflects the variability or spread of a sample statistic, such as a sample mean or proportion, from the true population parameter. Essentially, it quantifies the precision of the sample statistic as an estimate of the population parameter.
In formula terms, the standard error for a proportion is calculated as follows:\[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]where \(\hat{p}\) is the sample proportion, and \(n\) is the sample size.
It's important to note that the standard error decreases when the sample size increases, leading to more reliable and accurate estimates. Hence, collecting a larger sample can reduce the standard error, contrary to fewer data implying higher variability in estimates.

Poll Analysis

Poll analysis involves interpreting and evaluating the results obtained from surveys like the one described in the exercise. Such analysis requires an understanding of how to handle sample data, perform hypothesis testing, and calculate confidence intervals to make inferences about the population.
In poll analysis:

• The sample's representativeness of the population is crucial for result validity. A well-designed poll ensures that every member of the population has a chance of being selected.
• It's also important to consider the margin of error, determined by the standard error, which gives insight into potential variability or uncertainty in poll results.
• Analysts must critically assess the sample size's adequacy to ensure reliable estimates are produced.

Effective poll analysis allows us to understand population behaviors and preferences, provided the survey methodology is solid and the statistical interpretation sound.