Question

according to facebooks self - reported statistics, the average facebook user is connected to 80 community pages, groups, and events. for a statistics project a student at contra costa college tests the hypothesis that ccc students will average less than 80 such connections.
she posts a survey on her facebook page. her sample contains 45 responses.
she chooses a 5% level of significance. from her data she calculates a p - value of about 0.04.
what can she conclude?
○ nothing. the conditions for use of a z - model are not met. she cannot trust that the p - value is accurate for this reason.
○ the data is not statistically significant. in other words, the data do not provide enough evidence to conclude that the mean number of facebook connections to community pages, groups and events of all ccc college students is less the 80.
○ the data is statistically significant. in other words, the data do provide enough evidence to conclude that the mean number of facebook connections to community pages, groups and events of all ccc college students is less the 80.

Explanation:

1. First, recall the concept of p - value and significance level ($\alpha$). The significance level here is $\alpha = 0.05$ (5% level of significance), and the calculated p - value is 0.04.
2. The decision rule for hypothesis testing is: if the p - value $\leq\alpha$, we reject the null hypothesis. Here, $0.04\leq0.05$, so we reject the null hypothesis. The null hypothesis in this case is that the mean number of Facebook connections for CCC students is 80 (or more, depending on the alternative hypothesis), and the alternative hypothesis is that the mean is less than 80.
3. When we reject the null hypothesis at the given significance level, we can say that the data provides sufficient evidence (statistically significant) to conclude that the mean number of Facebook connections to community pages, groups, and events of all CCC college students is less than 80.
4. Now, let's analyze the options:

• Option 1: "Nothing. The conditions for use of a Z - model are not met. She cannot trust that the p - value is accurate for this reason." But in a hypothesis test for the mean, if the sample size $n = 45$ (which is relatively large, $n\geq30$), we can use the Z - model (by the Central Limit Theorem). So this option is incorrect.
• Option 2: "The data is not statistically significant. In other words, the data do not provide enough evidence to conclude that the mean number of Facebook connections to community pages, groups and events of all CCC college students is less the 80." Since $p - value=0.04\leq0.05=\alpha$, the data is statistically significant, so this option is incorrect.
• Option 3: "The data is statistically significant. In other words, the data do provide enough evidence to conclude that the mean number of Facebook connections to community pages, groups and events of all CCC college students is less the 80." This option is correct as we rejected the null hypothesis (that the mean is 80 or more) in favor of the alternative hypothesis (mean is less than 80) because the p - value is less than the significance level.

Answer:

The data is statistically significant. In other words, the data do provide enough evidence to conclude that the mean number of Facebook connections to community pages, groups and events of all CCC college students is less the 80.