Question

USA Today (Feb. 17, 2011) described a survey of 1,008 American adults. One question on the survey asked people if they had ever sent a love letter using e-mail. Suppose that this survey used a random sample of adults and that you want to decide if there is evidence that more than \(20 \%\) of American adults have written a love letter using e-mail. a. Describe the shape, center, and spread of the sampling distribution of \(\hat{p}\) for random samples of size 1,008 if the null hypothesis \(H_{0}: p=0.20\) is true. b. Based on your answer to Part (a), what sample proportion values would convince you that more than \(20 \%\) of adults have sent a love letter via e-mail?

Short answer

a) The shape of the sampling distribution of \(\hat{p}\) will be approximately normal (due to Central Limit Theorem), the centre (mean) would be 0.20 (the hypothesised population proportion), and the spread (standard deviation) computed from the formula. b) Values of \(\hat{p}\) that lead to a \(z\)-score greater than 1.645 (considering a significance level of 0.05) would convince that more than 20% of American adults have sent a love letter via e-mail.

Step-by-step solution

Determining parameters under the null hypothesis

Assuming the null hypothesis that \(p=0.20\) (20% of people have sent an e-mail love letter), the parameters of the sampling distribution can be computed. The mean of \(\hat{p}\) should be equal to \(p = 0.20\). The standard deviation of this distribution is given by the formula \(\sigma_{\hat{p}} = \sqrt{ \frac{p(1-p)}{n}}\), where \(p\) is the population proportion under the null hypothesis (0.20) and \(n\) is the sample size (1,008).

Compute standard deviation

Use the values in the formula to compute the standard deviation. When the values are plugged in, it becomes \(\sigma_{\hat{p}} = \sqrt{ \frac{0.20(1-0.20)}{1008}}\). Evaluating this expression gives the standard deviation of the sample proportion.

Describe sampling distribution

Using the Central Limit Theorem, it can be noted that the sampling distribution of \(\hat{p}\) will be approximately normal since the sample size is large (\(n >= 30\)). The center (mean) of the distribution is \(0.20\) and the spread (standard deviation) is what was computed in Step 2.

Proportion values to reject null hypothesis

To reject the null hypothesis (essentially believing that more than 20% of adults have sent a love letter via e-mail), the test statistic (\(z\)) is used, which is calculated as \( z = \frac{ \hat{p} - p }{ \sigma_{\hat{p}} }\). If \(z\) is significantly high, then the null hypothesis could be rejected. Note that the level of significance is not provided in the problem, so a common one like 0.05 could be assumed and a corresponding \(z\)-score of 1.645 for a one-tailed test could be used. So, values of \(\hat{p}\) that would lead to a \(z\)-score greater than 1.645 would convince that more than 20% of adults have sent an e-mail love letter.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about the properties of a population based on sample data. When we perform hypothesis testing, we use a process that starts with an assumption, known as the null hypothesis, and then determines whether the sample data provides sufficient evidence to reject this assumption in favor of an alternative hypothesis.

The process includes several steps: stating the null and alternative hypotheses, choosing a significance level, calculating a test statistic from the sample data, and then comparing this statistic to a critical value to decide whether to reject the null hypothesis. If the test statistic is beyond the critical value, the null hypothesis can be rejected, suggesting the sample data is significantly different from what was hypothesized.

Null Hypothesis

The null hypothesis, denoted as \( H_{0} \), is a statement made about a population parameter that indicates no effect or no change. It's essentially a skeptical stance or a claim of 'no difference'. For example, in the case of the love letter survey, the null hypothesis is that 20% or less of American adults have sent a love letter via email (\( H_{0}: p = 0.20 \)).

The null hypothesis serves as a baseline or a standard against which we compare the evidence provided by our sample data. If the evidence is strong enough, we might reject the null hypothesis in favor of the alternative hypothesis, which is the opposite claim—suggesting, in this context, that more than 20% of adults have indeed sent a love letter via email.

Standard Deviation

The standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of a sampling distribution, the standard deviation of the sample proportion (denoted as \( \sigma_{\hat{p}} \)) provides information about how much the sample proportions vary from the population proportion.

Calculating the standard deviation under the null hypothesis gives us an idea of how spread out the sampling distribution will be if the null hypothesis were true. This is essential when we perform hypothesis testing because it helps us understand how unusual or common our sample statistic is compared to the null hypothesis scenario.

Sample Proportion

The sample proportion (denoted as \( \hat{p} \)) represents the proportion of success cases in a sample. For instance, in the case of the love letter survey, \( \hat{p} \)) would be the proportion of the sample that reported having sent a love letter via email.

This is a key component in hypothesis testing because it is the actual value we compare to the hypothesized population proportion (\( p \)) under the null hypothesis. We use the sample proportion to calculate the test statistic, which we then use to make inferences about the entire population.

Central Limit Theorem

The Central Limit Theorem is a key statistical concept that states that the distribution of sample means (and by extension, sample proportions) will be approximately normally distributed, provided the sample size is large enough (usually \( n \geq 30 \) is considered sufficient).

For hypothesis testing, this theorem is crucial because it allows us to use the normal distribution as a model for our sampling distribution, even when we are unsure about the shape of the population distribution. This is the foundation for being able to use z-scores and look up probabilities in standard normal distribution tables, which is a common method of assessing the statistical significance in hypothesis testing.

Population Proportion

The population proportion (denoted as \( p \)) is the true proportion of the entire population that has a certain characteristic. It's what we are trying to make inferences about when we are conducting a survey.

For hypothesis testing, our goal is to determine whether the sample proportion \( \hat{p} \) from our sample is significantly different from the population proportion in a way that is not just due to random sampling variability. The population proportion is what we use to calculate the expected value and standard deviation of the sampling distribution under the null hypothesis and ultimately helps us decide whether our sample provides enough evidence to suggest a significant difference from this hypothesized proportion.