Own a Gun? In a Harris Poll conducted in May 2000 , 39 % of...

Own a Gun? In a Harris Poll conducted in May 2000 , $39 \%$ of the people polled answered yes to the following question: "Do you happen to have in your home or garage any guns or revolvers?" The margin of error in the poll was $\pm 3 \%=\pm 0.03,$ and the estimate was made with $95 \%$ confidence. How many people were surveyed?

Key Concepts

Confidence Interval

A confidence interval is a range of values used to estimate a population parameter (such as a proportion) with a certain level of confidence, typically expressed as a percentage. It reflects the uncertainty around the sample estimate and provides a boundary within which the true population parameter is expected to fall.

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate due to sampling variability. It is determined by a combination of the variability in the data, the sample size, and the desired confidence level.

Confidence Level and Z-Score

The confidence level indicates the probability that the calculated confidence interval will contain the true population parameter over many samples. For a 95% confidence level, the corresponding Z-score from the standard normal distribution (approximately 1.96) is used to calculate the margin of error and determine the sample size.

Sample Size Determination

Sample size determination involves calculating the number of observations required to achieve a specific margin of error for a given confidence level. The formula n = (Z² * p(1-p)) / (margin of error)² uses the Z-score, the estimated proportion (p), and the desired margin of error to compute the minimum sample size needed to ensure the reliability of the survey results.

Transcript

00:01 The following is a solution to number 24, and this looks at gun ownership.

00:03 The researchers asks, do you own a gun? and we are going to find the minimum sample size necessary if we want the margin of error to be within three percentage points of the actual population proportion of those that own a gun, if we want to be 95 % confident.

00:21 So i wrote this formula up here, n equals p hat times 1 minus p hat times z star over e quantity squared.

00:29 And the p hat is the sample proportion and then one minus p hat one minus the sample proportion and then this z star that comes from the degree of confidence that that's that critical value and then e stands for the margin of error so let's go ahead and look at there there actually aren't two parts so i don't need this a do i there we go okay so the margin of error like i said earlier was was three percentage points so 0 .03 that stands for 3 % and then the 95 % confidence if you don't have that z star that critical value memorized.

01:01 You really should.

01:02 It's 1 .96 whenever you're 95 degree, 95 % confident.

01:07 And i always tell my students there are really three, possibly four critical values that you really should commit to memory, the 90 % confidence interval, the 95 % confidence interval, the 98 and then the 99 % confidence interval.

01:19 Those are the most typical ones.

01:20 But if you don't know what that's fine, you can use your chart to find it.

01:23 It may take you a little bit longer, or you can use your calculator.

01:26 A nice thing about calculators, it does it for you so on a ti -84, that's what i have.

01:31 If you go to second and then var is where it says distribution, and then you go down to this inverse norm, and you put the area in as 0 .95, that represents the 95 % confidence interval.

01:41 The mean and standard deviation are always 0 and 1.

01:44 And then if you have the option of doing a tail, make it center.

01:48 Now, if you have an older version and you don't have the option of making, choosing the tail, then you would need to change this to 0 .975 as the area.

01:57 0 .975 or 0 .025.

01:59 One will give you positive, one will give you a navigative, but always use the positive.

02:03 But hopefully you have a newer version because it makes things easier.

02:06 Just going to call that center...

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