Question

A study \(^{54}\) shows that relationship status on Facebook matters to couples. The study included 58 college-age heterosexual couples who had been in a relationship for an average of 19 months. In 45 of the 58 couples, both partners reported being in a relationship on Facebook. In 31 of the 58 couples, both partners showed their dating partner in their Facebook profile picture. Men were somewhat more likely to include their partner in the picture than vice versa. However, the study states: "Females' indication that they are in a relationship was not as important to their male partners compared with how females felt about male partners indicating they are in a relationship." Using a population of college-age heterosexual couples who have been in a relationship for an average of 19 months: (a) A \(95 \%\) confidence interval for the proportion with both partners reporting being in a relationshipon Facebook is about 0.66 to 0.88 . What is the conclusion in a hypothesis test to see if the proportion is different from \(0.5 ?\) What significance level is being used? (b) A \(95 \%\) confidence interval for the proportion with both partners showing their dating partner in their Facebook profile picture is about 0.40 to \(0.66 .\) What is the conclusion in a hypothesis test to see if the proportion is different from \(0.5 ?\) What significance level is being used?

Short answer

(a) We can conclude that there is evidence to claim the proportion of couples who report being in relationship on Facebook is different from 0.5. The significance level used is 0.05. (b) There is no evidence to claim that the proportion of couples showing their partner in profile picture is different from 0.5, and the significance level used is 0.05.

Step-by-step solution

Analyzing Part (a)

A 95% confidence interval for the proportion of couples reporting being in a relationship on Facebook is given as 0.66 to 0.88. This interval helps us to make probabilistic statements about unknown quantities. The question asks to test the hypothesis that the true proportion is different from 0.5.

Hypothesis Testing Part (a)

To conduct the hypothesis test, we should check if the hypothesized value (0.5) falls within the given confidence interval (0.66 to 0.88). The hypothesized value does not fall within this interval, so we reject the null hypothesis. Therefore, there is evidence to claim that the proportion is different from 0.5.

Significance Level Part (a)

The confidence level was 95%. Since the significance level (\(\alpha\)) plus the confidence level must total 1, then the significance level is \(1 - confidence level = 1 - 0.95 = 0.05\). So, the significance level used is 0.05.

Analyzing Part (b)

A 95% confidence interval for the proportion of couples showing their dating partner in their Facebook profile picture is given as 0.40 to 0.66. Similar to part (a), this hypothesis test is to determine whether the true proportion is different from 0.5.

Hypothesis Testing Part (b)

We need to check if the hypothesized value (0.5) falls within the given confidence interval (0.40 to 0.66). In this case, the hypothesized value falls within this interval, so we do not reject the null hypothesis. Therefore, there is no evidence to claim that the proportion is different from 0.5.

Significance Level Part (b)

The confidence level used for this hypothesis test still remains 95%. Therefore, we use the same procedure to compute the significance level as in part (a). The significance level is \(1 - confidence level = 1 - 0.95 = 0.05\). Thus, the significance level used is 0.05.

Key concepts

Confidence Interval

The confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. In statistics, when we are trying to estimate some aspect of a population, like a mean or a proportion, we use a sample to calculate a point estimate and a confidence interval around that estimate. For example, the confidence interval given in the exercise for part (a) is 0.66 to 0.88, which tells us that we can be 95% confident that the true proportion of couples reporting being in a relationship on Facebook falls within this range.

It's essential to understand that the confidence level (95% in the exercise) signifies the frequency with which the calculated interval will contain the true parameter if the same population is sampled multiple times. The confidence interval provides a measure of precision for our estimate and is used for hypothesis testing, as it's related to the concept of statistical significance. If the confidence interval does not contain the value of the null hypothesis, we have grounds to reject the null hypothesis, as seen in part (a) of the exercise.

Null Hypothesis

The null hypothesis, denoted usually by \(H_0\), is a statement that there is no effect or no difference, and it serves as a starting point for statistical testing. In hypothesis testing, we collect data to assess whether the results are compatible with the null hypothesis or if there's sufficient evidence to support an alternative hypothesis \(H_1 or H_a\).

In the given exercise for both part (a) and part (b), we are testing the null hypothesis that the true proportion is 0.5, suggesting that there is no difference from a benchmark reference of 0.5. In hypothesis testing, if the value in question falls outside the confidence interval, like in part (a), this indicates that our sample provides enough evidence to reject the null hypothesis. However, if the value falls inside the confidence interval, as in part (b), we cannot reject the null hypothesis.

Significance Level

The significance level, denoted by \( \alpha \), is the probability of rejecting the null hypothesis when it is actually true—this is known as a Type I error. The significance level is a critical component in hypothesis testing, as it defines the threshold for how much chance we are willing to accept when drawing conclusions from our data.

In both parts (a) and (b) of the exercise, a 95% confidence level is used, which corresponds to a significance level of 5% (\(\alpha = 0.05\)). This means that there is a 5% risk of concluding that a difference exists when in fact there is no such difference. Setting a significance level is an arbitrary decision, but common choices include 0.01, 0.05, and 0.10. Deciding on the significance level depends on the context of the study and how much risk of error is acceptable.

Proportion Test

A proportion test is a type of statistical test used to determine whether there is a significant difference between the proportion of a certain outcome in a sample and a specific value of interest (often a population proportion or 50/50 split). In the exercise, a proportion test is used to assess whether the observed proportions of couples sharing relationship status or profile pictures on Facebook differ from 0.5.

In part (a), the proportion test leads us to reject the null hypothesis, suggesting a significant difference between the observed proportion and 0.5. Conversely, in part (b), the observed proportion falls within the confidence interval that includes 0.5, so we do not reject the null hypothesis. The proportion test thus helps determine if our observed sample proportions are likely reflective of the true population proportions or if they have occurred simply by random chance.