The following random sample was selected from a normal...

The following random sample was selected from a normal distribution: 4,6,3,5,9,3 . a. Construct a $90 \%$ confidence interval for the population $\operatorname{mean} \mu .$ b. Construct a $95 \%$ confidence interval for the population $\operatorname{mean} \mu .$ c. Construct a $99 \%$ confidence interval for the population $\operatorname{mean} \mu .$ d. Assume that the sample mean $\bar{x}$ and sample standard deviation $s$ remain exactly the same as those you just calculated, but that they are based on a sample of $n=25$ observations rather than $n=6$ observations. Repeat parts $\mathbf{a}-\mathbf{c}$. What is the effect of increasing the sample size on the width of the confidence intervals?

The following random sample was selected from a normal distribution: 4,6,3,5,9,3 .
a. Construct a $90 \%$ confidence interval for the population $\operatorname{mean} \mu .$
b. Construct a $95 \%$ confidence interval for the population $\operatorname{mean} \mu .$
c. Construct a $99 \%$ confidence interval for the population $\operatorname{mean} \mu .$
d. Assume that the sample mean $\bar{x}$ and sample standard deviation $s$ remain exactly the same as those you just calculated, but that they are based on a sample of $n=25$ observations rather than $n=6$ observations. Repeat parts $\mathbf{a}-\mathbf{c}$. What is the effect of increasing the sample size on the width of the confidence intervals?

Key Concepts

Confidence Interval

A confidence interval provides a range of values, derived from sample statistics, that is likely to contain the true value of an unknown population parameter. It is constructed based on a chosen confidence level, such as 90%, 95%, or 99%, reflecting the degree of certainty associated with the interval estimate.

t-Distribution

The t-distribution is used in statistical inference when the sample size is small or when the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which accounts for the additional uncertainty in estimating the population variance from a small sample.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available for estimating a parameter. In the context of constructing a confidence interval for the mean using the t-distribution, it is calculated as the sample size minus one, and it affects the shape of the t-distribution.

Margin of Error

The margin of error represents the maximum expected difference between the sample statistic and the true population parameter. It is calculated using the critical value from the t-distribution and the standard error of the sample mean, and it determines the width of the confidence interval.

Sample Size Impact

Increasing the sample size decreases the standard error, which in turn narrows the width of the confidence interval. This means that larger samples yield more precise estimates of the population parameter, as the uncertainty associated with the estimate is reduced.

Transcript

00:01 So you have a list of six numbers and you want to construct a 90 % confidence interval.

00:08 And so this will end up being the x bar plus or minus a t value with .05 in the upper tail and it will have five degrees of freedom since your sample size here is six.

00:22 And then we'll multiply that by the sample standard deviation divided by the square root of n which will be six.

00:28 And i'm going to use my software and make the t interval for this data and then we'll answer the question.

00:36 But that is the formula that you would use in order to find this.

00:43 And so the first interval for the 90 % confidence interval is 3 .12 to 6 .8 and it would round to 8.

00:55 Now on part b we want a 95 % confidence interval.

00:58 And the only thing that's going to change is that t value will become .025 and we'll have still five degrees of freedom.

01:07 And so stat, test, that t interval, changing that level of confidence to .95, that interval becomes 2 .61 to 7 .39.

01:26 And we see that as we increase the confidence level we make the sample, the width of the interval change and become larger.

01:35 So next one, the 99 % confidence interval, again, that will be a t value with .005 in the upper tail and still five degrees of freedom.

01:45 So stat, test, interval, t interval, and change that to .99.

01:52 And again, if you have to show that longhand you can go through and show those steps.

01:55 1 .25 out to 8 .75.

02:02 Now part d says assume that the sample mean and sample standard deviation remain the same as calculated above, however, we have a sample size of 25.

02:12 And so that will end up being that you'll end up having that x bar was 5 and you're going to have plus or minus and you're going to have the 2 .28 or the standard deviation, but now it's going to be divided by the square root of 25 rather than the square root of 6...

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