Death Penalty In a Harris Poll conducted in July 2000 , 64 %...

Death Penalty In a Harris Poll conducted in July 2000 , $64 \%$ of the people polled answered yes to the following question: "Do you believe in capital punishment, that is, the death penalty, or are you opposed to it?" The margin of error in the poll was $\pm 3 \%$ = ±0.03 , and the estimate was made with $95 \%$ confidence. How many people were surveyed?

Key Concepts

Margin of Error

The margin of error is the maximum expected difference between the true population parameter and the estimate derived from a sample. It quantifies the uncertainty associated with a sample estimate, indicating the range within which the true value is likely to lie.

Confidence Level

The confidence level represents the probability that the calculated confidence interval from a sample will contain the true population parameter. For instance, a 95% confidence level means that if the survey were repeated multiple times, about 95% of the resulting intervals would include the true value.

Critical Value

The critical value is a point on the standard normal distribution that corresponds with the desired confidence level, such as 1.96 for a 95% confidence level. It is used in the formula for the margin of error to determine how spread out the confidence interval is around the sample estimate.

Sample Size

Sample size is the number of individuals or observations included in the survey or study. It is determined by the desired margin of error, confidence level, and the estimated variability or proportion in the population. A larger sample size generally provides a more precise estimate of the population parameter.

Transcript

00:01 The following is the solution to number 23, and the researchers ask the question, do you agree with the death penalty? and we're going to find out how many are necessary, or the sample size necessary, in order to get a certain margin of error.

00:14 And that certain margin of errors within three percentage points of the actual population proportion.

00:19 So i wrote this formula up here, n equals p hat times one minus p hat.

00:23 So that's the sample proportion times one minus the sample proportion, times z star, that's the critical value divided by the margin of error quantity squared.

00:32 And that's the formula that's given in the book, but you can also derive that from the margin of error formula.

00:37 So we're going to use that to find the minimum sample size necessary in order to get within, in this case, three percentage points.

00:44 So that's actually given in the problem.

00:45 So the margin of error is 0 .03.

00:48 It's within three percentage points of the actual population proportion.

00:52 And 95 % confidence means that the z star is 1 .3.

00:57 Now if you don't have that one memorized, you really should.

01:00 That's probably the most common one.

01:02 I always tell my students 90, 95, and 99 for sure you need to have memorized and the 98 is also a good z score to have memorized as well.

01:12 But if you don't have it memorized, that's fine.

01:13 You can use your table certainly works or you can use a ti84 calculator, which is what i'm going to use to show you how i get that.

01:21 So if i go to second distribution, second bars, you're going to use this inverse norm down here so option three and then the area is going to be 0 .95 and that represents the middle 95%.

01:32 The mean is always zero and the standard deviation is always one for these standardized scores.

01:37 And then if you have the option to change the tail, it should be the center.

01:40 Now if you don't have the option then you have to change it up a little bit, you would have to actually put the area as 0 .025 or 0 .975...

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