a We are told that a national polling company asserts that 54 % of U.S. adults are married. We survey a random sample of 50 adults. We will make a conclusion about the accuracy of the following claim .
The population proportion is 0.54 when 31 adults in the survey are married in a random sample of 50 adults.
We will create a simulation by using a random number generator on a graphing calculator. We will choose 80 random samples of size 50. To generate the numbers we will press MATH. Then we will scroll to the right to the PRB menu and choose the fifth option, randInt(.
Now we will press ENTER. We want to select randomly 50 numbers from 0 to 99, and the numbers 1 through 54 will show adults who are married. Therefore, we will write 0, 99, and 50, respectively, after randInt(. Next, we will press ENTER to see the 50 random numbers.
When we scroll to the right we can see 50 generated numbers. Note that since the graphing calculator generates the numbers randomly, our numbers are just a sample. From here we will make a table and write the all generated numbers. We will also color the numbers 1 through 54.
When we count the numbers in the pink circles in the table, we can see 23 numbers have values between 1 and 54. In our case, this means that 23 out of 50 adults are married. Now we will find the sample proportion.
23/50=0.46
To analyze the claim, we want to simulate the case by choosing 80 random samples of 50 people. Therefore, we will repeat what we have done so far 80 times. The results of each sample are as follows.
With this information we will count the number of each proportion and draw a dot plot that shows the proportions of adults who are married.
Let's remember that we survey a random sample of 50 adults and 31 of them are married. Let's find its sample proportion.
31/50=0.62
Now we will look at how many time this result occurred in the simulation.
In the simulation 0.62 occurred 9 times in 80 samples. It is likely that 31 out of 50 adults are married, as the company asserts 54 % of the adults are married. Therefore, we can conclude that the claim of the company is probably accurate.
b Here we will make a conclusion about the accuracy of the claim that the population proportion is 0.54 when 19 adults in our survey are married. Therefore, we will find the sample proportion.
19/50=0.38
Now we will look the place of this proportion at the dot plot that we made in Part A.
In the simulation, this result did not occur. It is unlikely that 19 out of 50 adults are married, as the population proportion is 54 %. Therefore, we can conclude that the claim of the company is probably not accurate.
c In this sub-exercise we will assume that the true population proportion is 0.54. We will estimate the variation among sample proportions for a sample size of 50. To do this we will use the dot plot we made in Part A.
When we look at the dot plot we can see that it has a bell-shape. The shape is almost symmetric. Recall that in a normal distribution 95 % of the possible sample proportions will be within two standard deviations of 0.54. Since we have 80 sample proportions, we need to find 95 % of it.
80* 95 % &= 80* 95/100 [0.5em]
&= 76
We need to have 76 possible sample proportions. Therefore, we will exclude four numbers that have the least or the greatest sample proportions, which are the orange points.
Consequently, our 76 possible sample proportions should lie in the interval from 0.42 to 0.68.
