Question

The 2010 General Social Survey asked the question: "For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010.


(a) Interpret this interval in context of the data.

• 95% of surveys will report a mean number of ''not good'' days in the past 30 is between 3.40 and 4.24
• The researchers can be 95% confident that the true population mean number of ''not good'' days in the past 30 is between 3.40 and 4.24
• The researchers can be 95% confident that the sample mean number of ''not good'' days in the past 30 is between 3.40 and 4.24

(b) Suppose the researchers think a 99% confidence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% confidence interval?

• larger since the standard error would be larger
• smaller since we will be more sure of our results
• larger since the margin for error must be larger
• smaller since we have less room for error

Answer 1

A confidence interval :

It is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

a)

For this case we can conclude that we are 95% confident that the true mean for the variable of interest is between 3.4 and 4.24 days in the US population. The 95% confident means that if we select 100 different samples and calculate the 95% confidence interval for each sample selected, then we will have approximately 95 out of 100 confidence intervals will contain the true mean for the parameter of interest.

b)

If we have the same info but we want more confidence that implies that the confidence interval would be wider, since the margin of error increase with more confidence, because the critical value increase.