A giant gumball machine has only blue gumballs and red gumballs. Of the

Question

5000
gumballs in the machine,
2900
of them are blue and
2100
of them are red.

(a)Construct a percent bar graph for the gumballs in the machine.

50%50%
BlueRed

Let's simulate three random samples of
4
gumballs from the machine. Press all three "Run" buttons to do so. (You can press "Run" multiple times to generate more samples to see how they vary.)
Run
Run

Sample size:
0
Number
%
of totalBlue
Red
Run
Run

Sample size:
0
Number
%
of totalBlue
Red
Run
Run

Sample size:
0
Number
%
of totalBlue
Red
(b)In part (a), we calculated two different types of percentages: population percentages in the percent bar graph and sample percentages in the simulations.
The red population percentage is the percentage of the
5000
gumballs that are red.
The red sample percentage is the percentage of gumballs that are red in a random sample of
4
gumballs.
True or false?
StatementTrueFalseIn a random sample of
4
gumballs, the red sample percentage can be
50%
.
In a random sample of
4
gumballs, the red sample percentage can be equal to the red population percentage.
With a very large sample size, there must be no difference between the red sample percentage and the red population percentage.

Answer 1

Let's break down each part of your question regarding the gumball machine and the related statements.

Part (a): Percent Bar Graph for Gumballs

You provided the quantities of gumballs in the machine:

• Total Gumballs: 5000
• Blue Gumballs: 2900
• Red Gumballs: 2100

To calculate the percentages:

1. Blue Gumball Percentage: \[ \text{Percentage of Blue} = \left(\frac{2900}{5000}\right) \times 100 = 58% \]

2. Red Gumball Percentage: \[ \text{Percentage of Red} = \left(\frac{2100}{5000}\right) \times 100 = 42% \]

So the percent bar graph would show:

• Blue: 58%
• Red: 42%

Part (b): True or False Statements

Let's analyze each statement provided in part (b):

1. In a random sample of 4 gumballs, the red sample percentage can be 50%.

   • False. The maximum possible number of red gumballs in a sample of 4 would be 4 (which corresponds to a percentage of 100%), and the minimum would be 0 (corresponding to 0%). The probability of getting exactly 50% red gumballs (i.e., 2 red and 2 blue) is possible but not guaranteed. The specific statement must be carefully considered based on the actual draw.

2. In a random sample of 4 gumballs, the red sample percentage can be equal to the red population percentage.

   • True. While not guaranteed in every sample, it is possible for the sample percentage to match the population percentage, especially if the sample reflects the overall distribution of the gumballs.

3. With a very large sample size, there must be no difference between the red sample percentage and the red population percentage.

   • False. While larger sample sizes tend to provide estimates closer to the population parameters due to the law of large numbers, they do not guarantee that there will be no differences. There may still be sampling variability, and the sample percentage may vary somewhat from the population percentage, though the differences are expected to be minor with sufficiently large sample sizes.

In summary:

1. False
2. True
3. False