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Are You "In a Relationship"? A new study \(^{45}\) 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 relationship on 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?

Step by step solution

01

Evaluating the first confidence interval

This step focuses on assessing whether 0.5 could be a plausible value for the proportion of couples both reporting being in a relationship on Facebook. The 95% confidence interval is from 0.66 to 0.88. Since 0.5 does not fall into this range, we would conclude that, at a 95% confidence level, the proportion of couples both reporting being in a relationship on Facebook is significantly different from 0.5. This would lead us to reject the null hypothesis that the proportion is 0.5. The significance level associated with a 95% confidence level is 0.05.

02

Evaluating the second confidence interval

Following the same procedures as in Step 1, we turn our attention to the proportion of couples both exhibiting their dating partner in their profile picture on Facebook. The statement provides a 95% confidence interval from 0.40 to 0.66. Since 0.5 is within this range, it is a plausible value for the population parameter at a 95% confidence level. Therefore, we would not reject the null hypothesis that the proportion is 0.5 in this instance. The significance level for this test is also 0.05 as the confidence level is stated as 95%.

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Key Concepts

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

Confidence Interval

When we talk about confidence intervals in statistics, we are referring to the range within which we expect a certain population parameter to fall, with a specific level of certainty. Essentially, it is an educated guess about where an unknown population parameter (e.g., a mean or proportion) can lie based on a sample statistic.

For example, in our exercise regarding relationship statuses on Facebook, a 95% confidence interval for the proportion of couples both reporting being in a relationship is given as from 0.66 to 0.88. This interval was constructed from sampled data, and it suggests with 95% confidence that the true proportion of such couples in the entire population would fall within this range.

If we were to take many samples and calculate a range like this from each one, we'd expect the true population parameter to be within this range 95% of the time. The '95%' is what we call the 'confidence level' and directly reflects how sure we are that our interval contains the true parameter. Meanwhile, the 'confidence interval' offers a visual understanding of where the parameter lies, indicating, for instance, if a hypothesized value (like 0.5 in the exercise) is plausible or not.

Null Hypothesis

In hypothesis testing, the null hypothesis, often denoted as H0, is a statement suggesting there is no effect or no difference, and it serves as the starting point for statistical significance testing. It is the hypothesis that the researcher tries to disprove or reject.

In the context of the Facebook relationship status study, the null hypothesis might be that the true proportion of couples who display their relationship status or feature their partner in their profile picture is equal to 0.5 – essentially saying that it's equally likely as not. Hypothesis testing then assesses whether the sample data provide enough evidence to reject this hypothesis. If the confidence interval does not include the hypothesized value (0.5 in this case), like it doesn't in part (a) of the exercise, this suggests there is a significant difference, thus leading to the rejection of the null hypothesis.

Significance Level

The significance level, denoted by the Greek letter alpha (α), is the threshold used to determine whether a statistical result is not likely due to chance. This level represents the probability of rejecting the null hypothesis when it is in fact true, a scenario known as a 'Type I error'.

In most social science research, including our Facebook relationship status scenario, a common significance level used is 0.05, or 5%. This means there is a 5% risk of concluding that a difference exists when there is no actual difference. If the confidence interval excludes the hypothesized value, as it is in the (a) part of the Facebook study, and if we're using the 5% significance level, then we have strong evidence against the null hypothesis. Conversely, part (b) of the study demonstrates a scenario where the null hypothesis is not rejected, hence implying that there isn't strong evidence to suggest a difference from the hypothesized value, within the accepted risk of 0.05.