11.2.2.1 - Example: Summarized Data, Equal Proportions

Example: Tulips Section

A company selling tulip bulbs claims they have equal proportions of white, pink, and purple bulbs and that they fill customer orders by randomly selecting bulbs from the population of all of their bulbs.

You ordered 30 bulbs and received 16 white, 8 pink, and 6 purple.

Is there convincing evidence the bulbs you received were not randomly selected from a population with an equal proportion of each color?

Use Minitab to conduct a hypothesis test to address this research question. 

We'll go through each of the steps in the hypotheses test:

Step 1: Check assumptions and write hypotheses

\(H_0\colon p_{white}=p_{pink}=p_{purple}=\dfrac{1}{3}\)
\(H_a\colon\) at least one \(p_i\) is not \(\dfrac{1}{3}\)

We can use the null hypothesis to check the assumption that all expected counts are at least 5.

\(Expected\;count=n (p_i)\)

All \(p_i\) are \(\frac{1}{3}\). \(30(\frac{1}{3})=10\), thus this assumption is met and we can approximate the sampling distribution using the chi-square distribution.

Step 2: Compute the test statistic

Let's use Minitab to calculate this.

First, enter the summarized data into a Minitab Worksheet.

 

C1

C2

 

Color

Count

1

White

16

2

Pink

8

3

Purple

6

  1. After entering the data, select Stat > Tables > Chi-Square Goodness of Fit Test (One Variable)
  2. Double-click Count to enter it into the Observed Counts box
  3. Double-click Color to enter it into the Category names (optional) box
  4. Click OK

This should result in the following output:

Chi-Square Goodness-of-Fit Test: Count

Observed and Expected Counts

Category

Observed

Test
Proportion

Expected

Contribution
to Chi-Sq

White

16

0.333333

10

3.6

Pink

8

0.333333

10

0.4

Purple

6

0.333333

10

1.6

Chi-Square Test

N

DF

Chi-Sq

P-Value

30

2

5.6

0.061

The test statistic is a Chi-Square of 5.6.

Step 3: Determine the p-value

The p-value from the output is 0.061.
 

Step 4: Make a decision

\(p>0.05\) therefore we fail to reject the null hypothesis.

Step 5: State a "real world" conclusion

There is not enough evidence that the tulip bulbs were not randomly selected from a population with equal proportions of white, pink and purple.