A simple random sample of size n is drawn. The sample mean, x̅, is found to be 18.4, and the sam

A simple random sample of size n is drawn. The sample mean,...

A simple random sample of size $n$ is drawn. The sample mean, $\bar{x},$ is found to be $18.4,$ and the sample standard deviation, $s$, is found to be $4.5 .$ (a) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n,$ is 35 (b) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n,$ is $50 .$ How does increasing the sample size affect the margin of error, $E ?$ (c) Construct a $99 \%$ confidence interval about $\mu$ if the sample size, $n,$ is $35 .$ Compare the results to those obtained in part (a). How does increasing the level of confidence affect the margin of error, $E ?$ (d) If the sample size is $n=15,$ what conditions must be satisfied to compute the confidence interval?

A simple random sample of size $n$ is drawn. The sample mean, $\bar{x},$ is found to be $18.4,$ and the sample standard deviation, $s$, is found to be $4.5 .$
(a) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n,$ is 35
(b) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n,$ is $50 .$ How does increasing the sample size affect the margin of error, $E ?$
(c) Construct a $99 \%$ confidence interval about $\mu$ if the sample size, $n,$ is $35 .$ Compare the results to those obtained in part (a). How does increasing the level of confidence affect the margin of error, $E ?$
(d) If the sample size is $n=15,$ what conditions must be satisfied to compute the confidence interval?

Key Concepts

Assumptions for Confidence Interval Validity

When constructing confidence intervals, especially with small samples, it is important to ensure that the data are from a simple random sample, the observations are independent, and the population from which the sample is drawn is approximately normally distributed. These conditions help ensure the reliability of the interval estimation using the t-distribution.

Confidence Level

The confidence level indicates the degree of certainty that the calculated interval will capture the true population parameter. Higher confidence levels require a wider interval, meaning a larger margin of error, to ensure that the true value is included with the stated level of confidence.

Sample Size Effects

Increasing the sample size reduces the standard error of the mean, which in turn lowers the margin of error. This results in a more precise confidence interval. Conversely, a smaller sample size increases the margin of error, reflecting greater uncertainty in the estimate.

t-Distribution

The t-distribution is used when estimating the mean of a normally distributed population with an unknown population standard deviation, particularly when sample sizes are small. This distribution accounts for additional uncertainty by having heavier tails than the normal distribution, and it depends on the degrees of freedom, which are related to the sample size.

Confidence Interval

A confidence interval provides a range of plausible values for an unknown parameter, such as a population mean, based on sample estimates. It combines the sample statistic, a measure of variability (like the standard deviation), and a critical value from a probability distribution to quantify the uncertainty associated with the estimate.

Margin of Error

The margin of error measures the maximum expected difference between the true population parameter and the sample estimate within a given confidence level. It is influenced by the critical value (from the t- or z-distribution), the sample standard deviation, and the sample size; larger sample sizes typically yield a smaller margin of error, indicating more precise estimates.

Transcript

00:01 The following is the solution to number 11, and we're given summary stats that the sample mean x bar is 18 .4, and the sample standard deviation is 4 .5.

00:09 Now, we only know the sample standard deviation, not the population standard deviation.

00:14 So since we don't know the population standard deviation sigma, we have to use the t interval instead of the z interval.

00:20 So keep that in mind moving forward.

00:22 So i'm going to use technology for this problem here.

00:25 We basically have three questions that are very, very similar, and then one where we talk a little.

00:30 Little bit about conditions for inference.

00:32 Now it doesn't say anything about the normality of this population.

00:35 That's going to come up here in part d, but it doesn't matter because these sample sizes are big.

00:40 They're greater than 30.

00:41 So it doesn't matter how this population is distributed.

00:45 That sample size is big enough.

00:47 So we're asked to find the 95 % confidence interval whenever n is 35.

00:51 Then we find the 95 % confidence interval whenever end is 50, and we're going to compare that.

00:55 So the same confidence level, but the sample size is a little bit bigger.

00:59 Then we come down here.

01:00 Here and we increase that confidence level to 99 % confidence.

01:05 And then we're back down to the n equals 35.

01:07 So let's take a look and see what happens.

01:11 So i'm going to use the ti84, but you can use any sort of technology you want or you can go and do the formula.

01:16 But if you go to stat and then arrow over to tests, and it's this eighth option here, the t interval.

01:23 And make sure that the summary stats is highlighted because we don't have data, we're just given summary stats.

01:28 And x bar was 18 .4.

01:29 The s was 4 .5.

01:31 The n for this first one was 35, and then that confidence level is 0 .95, and that represents 95%.

01:37 So whenever we calculate, this top band here, that's our confidence level.

01:42 So about 16 .9 to 19 .9 is going to be our interval.

01:47 So let's go ahead and write that down.

01:48 So 16 point, i'll go ahead and do what they did, so 854, and then the upper bound is 19 .946.

01:59 So that's our first confidence interval.

02:00 Now, let's see what happens whenever we increase that sample size from 35 to 50...

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