a We are told that a national polling company asserts that only 29 % of U.S employees feel engaged at work. We conduct a survey of a random sample of 50 adults, and we will make a conclusion about the accuracy of the following claim .
Therefore, we will create a simulation by using a random number generator on 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(.
Then we will press ENTER. We want to select randomly 50 numbers from 0 to 99, and the numbers 1 through 29 will show the employees who feel engaged at work. Therefore, we will write 0, 99, and 50, respectively after randInt(. Next, we will press ENTER to see the 50 random numbers.
If we scroll to the right we can see 50 generated numbers. Note that since the graphing calculator generates the numbers randomly, the generated set of numbers gives us a sample set. From here we will make a table and write the all generated numbers. Also, we will highlight the numbers 1 through 29.
In the table the numbers in the pink circles give us the number of employees who feel engaged at work, which is 16 out of 50 employees. Now we will find the sample proportion.
16/50=0.32
Next, to analyze the claim we want to simulate the case by choosing 80 random samples of 50 people. Therefore, we will proceed in the same way 80 times. The result of each sample is as follows.
By using this information we will count the number of each proportion and draw a dot plot that shows the proportions of employees who feel engaged at work.
Let's recall that in our claim we survey a random sample of 50 adults and 16 of them feel engaged at work. Let's find its sample proportion.
16/50=0.32
From here we will look at how many time this result occurred in the simulation.
In the simulation 0.32 occurred 7 times in 80 samples. It is likely that 16 out of 50 employees feel engaged at work, as the company asserts 29 % of the employees feel engaged at work. 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 0.29 when 23 adults in our survey are married. Therefore, we will find the sample proportion.
23/50=0.46
Now we will look the place of this proportion in the dot plot that we made in Part A.
In the simulation this result did not occur, so it is unlikely that 23 out of 50 adults are married, as the population proportion is 29 %. Therefore, we can conclude that the claim of the company is probably not accurate.
c This time we will assume that the true population proportion is 0.29. We will estimate the variation among sample proportions for sample size 50. To do this, one more time we will use the dot plot we made in Part A.
From the dot plot, we can say that it has a quite symmetric bell-shape. Recall that in a normal distribution 95 % of the possible sample proportions will be within two standard deviations of 0.29. 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 the four numbers that have the least or the greatest sample proportions, which are selected as the orange points.
Consequently, our 76 possible sample proportions should lie in the interval from 0.14 to 0.38.
