Answer:
II and III
See explanation above
Step-by-step explanation:
When all other rings remain the same, which of the following conditions would have resulted in a wider interval than the one constructed?
I. A sample size of 20 with 95 percent confidence
II. A sample size of 15 with 99 percent of confidence
III. A sample size of 12 with 95 percent confidence
Previous concepts
A confidence interval 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".
represent the sample mean
population mean (variable of interest)
s represent the sample standard deviation
n represent the sample size
The confidence interval for the mean is given by the following formula:
The margin of error is given by:
I. A sample size of 20 with 95 percent confidence
The original sample size was 15, the degrees of freedom for the original interval would be n-1=14 , the value for and and then the critical value is
And then the Margin of error would be:
For the new sample size n=20 the degrees of freedom for the new interval interval would be n-1=19 , the value for and and then the critical value is
And then the Margin error would be:
So then we have a lower margin of error so then we will have a shorter interval
II. A sample size of 15 with 99 percent of confidence
The original sample size was 15, the degrees of freedom for the original interval would be n-1=14 , the value for and and then the critical value is
And then the Margin of error would be:
For the new interval the sample size n=15 the degrees of freedom for the new interval interval would be n-1=14 , the value for and and then the critical value is
And then the Margin error would be:
So then we have a greater margin of error so then we will have a wider interval. And this makes sense since with more confidence the interval needs to be wider.
III. A sample size of 12 with 95 percent confidence
The original sample size was 15, the degrees of freedom for the original interval would be n-1=14 , the value for and and then the critical value is
And then the Margin of error would be:
For the new interval the sample size n=12 the degrees of freedom for the new interval interval would be n-1=11 , the value for and and then the critical value is
And then the Margin error would be:
So then we have a greater margin of error so then we will have a wider interval.