You construct three 88% confidence intervals as follows:
A. A t-interval with 6 degrees of freedom.
B. A t-interval with 2 degrees of freedom.
C. A z-interval
Assuming the mean and standard deviation are the same for all three intervals, write the three intervals (A, B, and C) in order, from narrowest to widest.
here is my work:
c. z interval, a t-interval df 6, b. t interval df 2
Is this correct?
Well, let's see here. Confidence intervals are like clothes for data, and just like clothes, some can be tighter or looser than others. Now, let's line up these intervals from tightest to loosest:
C. The z-interval, my friend, is the tightest of the bunch. It always thinks it's better than the rest with its fancy z-statistic. It's all, "Look at me and my known standard deviation!" Hmph.
A. Now, we move on to the t-interval with 6 degrees of freedom. It's like someone squeezed it with a tight pair of pants. It's somewhat narrower than the others, but still has room to breathe.
B. Finally, we come to the t-interval with 2 degrees of freedom. Oh boy, this poor interval is like a clown trying to fit into a tiny car. It's the widest of them all, struggling to contain all that uncertainty.
So, the order from narrowest to widest is: C, A, B. Just like a line of clowns trying to squeeze through a door!
To compare the three intervals from narrowest to widest, we need to understand that the width of a confidence interval is determined by the critical value and the sample size.
A. A t-interval with 6 degrees of freedom:
Since the degrees of freedom are low, the critical value for this interval will be higher, resulting in a wider interval.
B. A t-interval with 2 degrees of freedom:
With even lower degrees of freedom, the critical value will be even higher, resulting in an even wider interval.
C. A z-interval:
For a z-interval, we use a standard normal distribution, which has more degrees of freedom compared to a t-distribution. Therefore, the critical value will be lower, resulting in a narrower interval.
From narrowest to widest, the order of the intervals would be:
C (z-interval), A (t-interval with 6 degrees of freedom), B (t-interval with 2 degrees of freedom).
To compare the widths of the three confidence intervals, we need to understand how the degrees of freedom and the type of interval (t or z) affect the interval width.
Degree of freedom (df) is a parameter that determines the shape of the t-distribution. A higher df leads to a distribution that is closer to a standard normal distribution (z-distribution).
The t-interval takes into account the uncertainty in estimating the population standard deviation, which increases with smaller sample sizes. The z-interval assumes that the population standard deviation is known.
Now, let's construct the three intervals and determine their widths:
A. A t-interval with 6 degrees of freedom:
To construct a confidence interval, you would typically need the sample mean, sample standard deviation, and sample size. However, since only the degrees of freedom are given, we cannot determine the exact width. Generally, a t-interval with a higher degree of freedom (like 6) will be narrower compared to a t-interval with a lower degree of freedom (like 2).
B. A t-interval with 2 degrees of freedom:
Similarly, without the necessary information like sample statistics, we cannot calculate the exact width of this interval. However, since the degrees of freedom are lower (2), the interval is likely to be wider than the t-interval with 6 degrees of freedom.
C. A z-interval:
A z-interval assumes that the population standard deviation is known, which implies that the sample size is large enough to accurately estimate it. Since there are no specific degrees of freedom mentioned, we can assume that the sample size is large. A large sample size results in a smaller standard error, making the z-interval narrower compared to the t-intervals.
Therefore, from narrowest to widest, the order of the three intervals is: C (z-interval) < A (t-interval with 6 degrees of freedom) < B (t-interval with 2 degrees of freedom).