A simple random sample of size n is drawn from a population...

A simple random sample of size $n$ is drawn from a population that is normally distributed. The sample mean, $\bar{x},$ is found to be $108,$ and the sample standard deviation, $s,$ is found to be $10 .$ (a) Construct a $96 \%$ confidence interval about $\mu$ if the sample size, $n,$ is 25 (b) Construct a $96 \%$ confidence interval about $\mu$ if the sample size, $n,$ is $10 .$ How does decreasing the sample size affect the margin of error, $E ?$ (c) Construct a $90 \%$ confidence interval if the sample size, $n,$ is $25 .$ Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the size of the margin of error, $E ?$ (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why?

A simple random sample of size $n$ is drawn from a population that is normally distributed. The sample mean, $\bar{x},$ is found to be $108,$ and the sample standard deviation, $s,$ is found to be $10 .$
(a) Construct a $96 \%$ confidence interval about $\mu$ if the sample size, $n,$ is 25
(b) Construct a $96 \%$ confidence interval about $\mu$ if the sample size, $n,$ is $10 .$ How does decreasing the sample size affect the margin of error, $E ?$
(c) Construct a $90 \%$ confidence interval if the sample size, $n,$ is $25 .$ Compare the results to those obtained in
part (a). How does decreasing the level of confidence affect the size of the margin of error, $E ?$
(d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why?

Key Concepts

Effect of Confidence Level

The level of confidence used in constructing an interval affects the critical value from the t-distribution: higher confidence levels require larger critical values, thus increasing the margin of error and yielding a wider interval. Lowering the confidence level results in a smaller critical value, which reduces the margin of error and leads to a narrower interval.

Assumption of Normality

When constructing confidence intervals using the t-distribution, it is crucial that the underlying population is normally distributed, especially for small samples. This assumption ensures that the sampling distribution of the sample mean is appropriately modeled by the t-distribution. For larger samples, the Central Limit Theorem can justify the use of the t-distribution even when the population is not perfectly normal.

Effect of Sample Size

The sample size directly influences the standard error—the larger the sample size, the smaller the standard error, leading to a smaller margin of error. Conversely, a smaller sample size increases the standard error, thereby widening the confidence interval. This concept highlights the trade-off between the precision of an estimate and the number of observations taken.

Confidence Interval

A confidence interval is a range of values used to estimate an unknown population parameter with a certain level of confidence. In the context of estimating the mean, the interval is constructed symmetrically around the sample mean and incorporates both the inherent variability in the data and the selected confidence level, which reflects the degree of certainty one has that the interval contains the true parameter.

Margin of Error

The margin of error quantifies the extent of the uncertainty associated with a sample estimate. It is calculated as the product of a critical value (derived from a t-distribution when population standard deviation is unknown) and the standard error (which is the sample standard deviation divided by the square root of the sample size). A larger margin of error indicates a wider interval and less precision in the estimation of the population mean.

t-distribution

The t-distribution is a probability distribution used when making inferences about a population mean when the sample size is small and the population standard deviation is unknown. It accounts for extra variability introduced by estimating the standard deviation from the sample, and its shape depends on the degrees of freedom, which are typically related to the sample size.

Transcript

00:01 The following is the solution to number nine, and this we're given the sample mean x bar is 108, and the sample standard deviation, not the population standard deviation, the sample standard deviation is 10.

00:10 And we're asked to find the 96 % confidence interval whenever n equals 25.

00:14 Now, since i only know s, i only know the sample standard deviation, i need to use the t interval instead of the z interval.

00:21 So you can do manually, i'm going to use the t i84 because it does it really nice.

00:26 So if you go to stat and then tests, it's going to be this eighth option.

00:30 Down here the t interval.

00:31 So usually if you know sigma you're going to use the z interval but this time i'm going to go to the eighth option the t interval.

00:38 Oops, do that again so stat tests and then eight.

00:42 Okay and then make sure you're hovered over stats here or that's highlighted and then you can start typing in your summary stats so x bar is 108 s is 10 and then the sample size was 25 and then the c level is 0 .96 and whenever we calculate that this first line here that's going to be the confidence intervals.

01:02 So 103 .66 all the way up to 112 .34.

01:07 Okay, so 103 .66 and 112 .34.

01:16 Okay, now we're going to do the 96 % confidence interval whenever we decrease the sample size to 10.

01:21 So let's see what happens here.

01:23 If we go back to stat, and you can probably guess what's going to happen, and then test, and then we go down to this eighth option again, and then everything stays the same except this time the n equals.

01:32 10 and everything else stays the same.

01:36 So we calculate and that gives us this confidence interval, 100 .42 all the way up to 115 .58.

01:45 So let's write that down and compare 100 .42 all the way up to 115 .58.

01:54 Okay, so what's the relationship between n and the margin of error or the interval size? well, as you can see as n decreases the margin of air, is going to increase making the confidence interval wider.

02:18 And this is super typical.

02:20 So we've seen this before with the z interval.

02:23 The less you know or the smaller your sample size, the more variable that's going to be, especially with the t distribution...

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