Determine the point estimate of the population mean and margin of error for each confidence inte

Determine the point estimate of the population mean and...

Key Concepts

Effect of Confidence Level on Margin of Error

Increasing the confidence level results in a larger critical value, which widens the confidence interval. This means that, while a higher confidence level increases the probability that the interval contains the population mean, it also increases the margin of error, reflecting greater uncertainty in the precision of the estimate.

Conditions for Computing Confidence Intervals

To validly compute a confidence interval, especially with small sample sizes, the key assumptions include that the data are drawn from a normally distributed population or the sample size is large enough for the Central Limit Theorem to apply, that the sample is a simple random sample from the population, and that the observations are independent. When dealing with small samples, additional caution is needed to verify normality.

Effect of Sample Size on Margin of Error

Increasing the sample size decreases the standard error, which in turn reduces the margin of error. A larger sample provides more reliable information about the population, leading to a narrower confidence interval and a more precise estimate of the population mean.

Point Estimate

A point estimate is a single value used to estimate a population parameter, such as the population mean. In the context of estimating a population mean, the sample mean is typically used as the best point estimate because it is an unbiased estimator of the true mean.

Margin of Error

The margin of error quantifies the extent of the uncertainty in the estimation of the population parameter. It is added to and subtracted from the point estimate to form the confidence interval and depends on the critical value from the relevant statistical distribution, the sample standard deviation, and the sample size.

Critical Value

The critical value is obtained from a statistical distribution (z-distribution for large samples or known population standard deviation, and t-distribution when the sample size is small or the population standard deviation is unknown) corresponding to the desired confidence level. It determines the width of the confidence interval by scaling the standard error.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true value of the population parameter with a certain level of confidence. It provides an estimate along with a measure of uncertainty associated with the sampling process.

Transcript

00:04 In this video, we're going to look at how to find a confidence interval for a population mean when we have a large sample size.

00:13 So here we have a random sample was taken from a population, and we want to find the confidence interval, 90 % confidence interval for the population mean if my sample size n equal 40.

00:27 And in my sample, i had a sample mean of 35 .1 and a sample standard deviation of 8 .3.

00:34 Now since my sample size is greater than 30, i can form my confidence interval for my population mean by the lower bound being my point estimate x bar minus the margin of error and the margin of error is found by tc alpha over 2 times s over the square to n and the upper bound is my point estimate from you x bar plus my margin of error and that's t sub alpha over two times s over the square of n now here the point estimate x bar is given to us in the question it's x bar is equal to three point or 35 .1 and for the margin of error we need to find our critical value t sub alpha over two so t sub alpha over two work recall that we are going to take the competence level 90.

01:38 Percent in decimal form .90, subtract that from 1 for alpha.

01:44 So alpha is 1 minus .90 or 0 .10.

01:49 Then alpha over 2 is we divide that .10 by 2.

01:53 I get 0 .05.

01:55 And then looking up on the table where i have my degrees of freedom for the t distribution i need a degrees of freedom, degrees of freedom is n minus 1.

02:05 So here my n is 40.

02:08 So 40 minus 1 gives me degrees of freedom is equal to 39.

02:13 Now when i use a table, if i look under the column heading of 0 .05, if my table is set up, so it gives you the area and the right tail as the column headings.

02:24 And my row heading is degrees of freedom across from 39.

02:28 You will get that the tsa alpha over 2 for this setting is the 1 .685.

02:35 So 1 .685.

02:38 On a calculator, we could go to our distribution.

02:45 So we go second distribution and then go down to the inverse t.

02:52 And for the area, what it wants us to put in is the area to the left of that right tail.

03:01 So 0 .05 was in my right tail.

03:05 1 minus 0 .05 is a .95.

03:08 So i'm going to enter 0 .95.

03:10 My degrees of freedom is 39.

03:16 And when i cursor down and paste it, and then push enter to have it work it out, you'll see the 1 .648, rounded is 1 .685.

03:30 So same value for your critical value.

03:33 So now when we plug in our values, x bar is 35 .1 and then minus my tsa b alpha over 2 is 1 .685 times my standard deviation sample standard deviation is 8 .7 and that is divided by the square root and my sample size is 40.

04:01 That's the lower bound and then the upper bound i have the 35 .1.

04:09 And then plus the 1 .685 times the s, which is my sample standard deviation 8 .7, and then divided by the square root of my sample size, the square root of 40.

04:28 Now, when we go through and calculate this margin of error, which is what is subtracted off of the point estimate and added to the point estimate for confidence intervals for the population mean, i have a 35 .1 minus a 2 .32.

04:51 And then the upper level is 35 .1 plus the 2 .32.

04:59 And then that gives me a 32 .78 as a lower bound and a 37 .42 as an upper bound.

05:11 So if we were to write this out in a sentence, we would say a 9 .7 .42 as an upper bound.

05:14 90 % confidence interval for the population mean is between 32 .78 and 37 .42.

05:26 Now let's look at how we could do this on the calculator.

05:28 If you're allowed to use a graphing calculator, you're going to push the stat button, and then cursor right to tests, and then cursor down to t intervals.

05:39 So we're doing confidence intervals, and the critical value is the student's t -difference.

05:45 Distribution, so you go to t interval.

05:48 And then if you add all the numbers from your sample, that would be data.

05:54 But they told us the mean and center deviation.

05:56 So that's stats.

05:58 My x bar for this question is 35 .1.

06:03 My s for this question is 8 .7.

06:07 And then my sample size is 40 with a confidence level of 0 .99, or 0 .90, excuse me.

06:17 And then when we calculate this, we see our 32 .78 as our lower bound and our 37 .42 as our upper bound.

06:29 So if you're in a class where you can only use a scientific calculator and formulas, you want to do it by how it was written out on the whiteboard.

06:37 If you're able to do this with your calculator, you can go ahead and calculate it in the stats menu.

06:44 Now, one thing i want to say is if they ask you for a point estimate, and the margin of error and you did it on your calculator, your point estimate is your x bar.

06:55 So your point estimate would still be 35 .1.

06:58 And you can find your margin of error then just by taking the upper bound of the 37 .418 and subtracting that 35 .1.

07:08 And that would give you that your margin of error there is the 2 .32 that we got for it.

07:15 Okay, let's look at the next part.

07:17 B...

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