Assume simple random sampling. Calculate the 95 % confidence...

Assume simple random sampling. Calculate the $95 \%$ confidence interval estimate for the population mean for each of the following.
a. $N=1200 ; n=80 ; s=10 ; \bar{x}=142$
b. $N=1425 ; n=90 ; s^2=64 ; \bar{x}=232.4$
c. $N=3200 ; n=200 ; s^2=129 ; \bar{x}=59.3$

Key Concepts

t-distribution

The t-distribution is commonly used in constructing confidence intervals for the population mean when the population standard deviation is unknown and the sample size is relatively small. It accounts for additional uncertainty in the estimate of the standard error compared to the normal (z) distribution.

Finite Population Correction Factor

When sampling without replacement from a finite population, the finite population correction factor is used to adjust the standard error to account for the reduced variability in the sample. This correction becomes important as the sample size becomes a significant fraction of the total population.

Standard Error

The standard error measures the variability or spread of the sampling distribution of a statistic, such as the sample mean. It quantifies the extent to which the sample mean would vary from sample to sample, and it is essential for constructing confidence intervals.

Confidence Interval

A confidence interval provides a range of plausible values for a population parameter based on sample data. It reflects the uncertainty inherent in estimating the parameter, and the chosen confidence level (e.g., 95%) means that, in repeated sampling, the interval will contain the true parameter value a specified proportion of the time.

Simple Random Sampling

Simple random sampling is a sampling method where every member of the population has an equal and independent chance of being selected. This method ensures that the sample is unbiased and representative of the population, making it a fundamental concept in inferential statistics.

Transcript

00:01 So we want to find the 95 % confidence interval for three problems.

00:04 We actually had this data in the previous question.

00:07 And we have to use this little finite population correction factor.

00:15 And so our first one, we have our sample size is, well, our population size is 1 ,200.

00:25 Our sample size is 80, and that's more than 5 % of the population.

00:29 So that's why we have to use this factor.

00:33 And our s value, our standard deviation is 10.

00:36 Our x bar is 142.

00:38 So our confidence interval for 95 % will be 142.

00:43 Now we need that t value with 0 .025 in the lower tail and 0 .025 in the upper tail with 79 degrees of freedom.

00:52 And then we have our sample standard deviation, which is 10, divided by the square root of 80.

00:58 And then we have to multiply it by that factor of 1200 minus the sample size of 80, so that will be 1120, and then we'll have 1199.

01:10 And when we do that calculation, we find that our lower limit is 139 .95, and then i'm going to write it as an inequality, and then i'll change that to an addition sign.

01:31 And that addition sign, we end up having 144 .15.

01:39 Now on part b, again, this data was used in the previous one.

01:43 And we'll just start writing down our confidence interval.

01:46 We'll have our mean, which is 232 .4.

01:50 Now our t value needs to have, for 95 % confidence, needs to have 89 degrees of freedom...

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