math help (project)

MAT 300

M&Ms® Project

Part 3 (21

p

ts)
We will be co

n

structing confidence intervals for the proportion of each color as well as the mean number of candies per bag. You will use the methods of 6.3 for the proportions and 6.1 for the mean. For the Bonus, you will use the sample si

z

e formula on page 338.

You can use StatCrunch to assist with the calculations. A link for StatCrunch can be found under Tools for Success in Course Home. Here is also a link: http://statcrunch.pearsoncmg.com/statcrunch/larson_les4e/dataset/index.html. You can also find additional help on both confidence intervals and StatCrunch in the Online Math Workshop under Tab: “MAT300 Archived Workshops”. Specifically you will be looking for Confidence Intervals and Using Technology – CI.

Submit your answers in

E

xcel, Word or pdf format. Submit your file through the M&M® project link in the weekly course content. If calculating by hand, be sure to keep at least 4-6 decimal places for the sample proportions to eliminate large rounding errors.

3 pts. Construct a 95% Confidence Interval for the proportion of blue M&Ms® candies.

3 pts. Construct a 95% Confidence Interval for the proportion of orange M&Ms® candies.

3 pts. Construct a 95% Confidence Interval for the proportion of green M&Ms® candies.

3 pts. Construct a 95% Confidence Interval for the proportion of yellow M&Ms® candies.

3 pts. Construct a 95% Confidence Interval for the proportion of red M&Ms® candies.

3 pts. Construct a 95% Confidence Interval for the proportion of brown M&Ms® candies.

3 pts. Construct a 95% Confidence Interval for the mean total number of candies (large samples).

BONUS:  5 pts.   How many candies should be sampled to obtain a 95% CI of the proportion of blue candies with a 4% margin of error if the known proportion of blue candies is 0.24?

HELP:

Color Proportions

You will need the information from the Part 2 Summary. For the colors, the confidence intervals need to be found using the formulas in section 6-3 for proportions. The margin of error formula is

n

q

p
z
E

ˆ

ˆ

=

p
ˆ
is the sample proportion of the color. It will change for each color. 

q
ˆ
is found by 1 –
p
ˆ
, so it will also change for each color. 
n is the total number of candies sampled. 
So let us do an example with “purple”. Let’s say there were 732 purple candies out of 3500 total candies. The sample proportion of purple candies is 732/3500 = 0.209143. This is what you did in part 2. Now to find the confidence interval, we need to calculate E. Let’s construct a 95% CI. For that confidence level (95), the z-value is 1.96. We also need

q
ˆ
:
q
ˆ
= 1 –
p
ˆ
= 1 – 0.209143 = 0.790857. 
Now let’s plug in:

013474
.
0
3500
790857
.
0
209143
.
0
96
.
1
=
×
=
E

The confidence interval is found by
p
ˆ
– E,
p
ˆ
+ E 

p
ˆ
– E: 0.209143 – 0.013474 = 0.19567 = 19.57% 

p
ˆ
+ E: 0.209143 + 0.013474 = 0.22262 = 22.26% 
So the confidence interval is (0.1957, 0.2226). 
You will follow this procedure for EACH color.
IF you have a TI 83/84, you can do the following: STAT, TESTS, 1-PropZInt, ENTER
NOTE: 1-PropZInt: 1 proportion z confidence interval
x = total number of that color
n = total number of candies
Then on the next screen enter 732 next to x, 3500 next to n, 0.95 next to C-Level and then calculate enter.
On the next screen the second line is the confidence interval: (0.19567,0.22262)
The third line is the
p
ˆ
value: .2091428571
On the last line is the sample size: 3500
IF you want to use StatCrunch, you will select Stat > Proportions > one sample > with summary. Then it will ask for the number of successes (total number of that color) and number of observations (total number of candies). Click on Next. In the next screen, click on the radio button to the left of “Confidence Interval”, enter the decimal of the confidence level and then click Calculate. The output will have the confidence interval:

95% confidence interval results:
p : proportion of successes for population
Method: Standard-Wald

Proportion

Count

Total

Sample Prop.

Std. Err.

L. Limit

U. Limit

P

732

3500

0.20914286

0.006874427

0.19566923

0.2226165

Mean

To find the confidence interval for the mean number of candies, you will need
x
(sample mean), s (sample standard deviation) and n. All of these values were summarized on Part 2 Summary. Here n is the number of BAGS sampled. The margin of error formula is
n
s
z
E
×
=

IF using the TI 83/84: STAT, TESTS, ZInterval, Enter.
Select Inpt: Stats
s: enter s value to at least 4 decimal places

x
: enter
x
value to at least 4 decimal places
n: enter number of bags sampled
C-Level: enter confidence level desired
Calculate, ENTER
On the next screen the second line is the confidence interval
Then
x
and n.
IF using StatCrunch, it is best if you copy the num. candies in bag data into StatCrunch, but you can use the summary information. The path is Stat > Z Statistics > one sample > with summary (if using summary information) or with data (if data is entered into the column).
If using summary: You will be prompted for the sample mean, sample standard deviation and sample size. Click Next. In the next screen, click the button to the left of Confidence Interval and then enter the confidence level as a decimal and then click Calculate.
If using data: You will first select the column with the data, then click next. The next screen is the same as with summary.
At the end of this project, you will be writing a report, explaining the method and presenting the results from each part of the project. You might find it useful to write this as you complete the work, so the report will be mostly written by the time it is assigned.

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