A flower shop wishes to add the valuable Waimea orchid to its product list. They purchase a large shipment of bulbs from a supplier in Kauai. It is established by Mendelian theory that the predominant colors in the Waimea orchid (blue, red, violet, orange) will occur with the ratio of 6:4:3:2. When the first 60 Waimea orchid bulbs bloom, the predominant colors are 27 blue, 10 red, 17 violet, 6 orange. The florists are concerned that these bulbs are not Waimea orchids, but a similar appearing (and more common, less valuable) hybrid. What hypothesis test can be used to test this? Conduct the test at α = 0.05 using Minitab. Interpret your results and state your conclusion in terms of the data.
To test whether the bulbs purchased by the flower shop are true Waimea orchids or a similar hybrid, you can use a hypothesis test known as the chi-square goodness-of-fit test.
The chi-square goodness-of-fit test compares the observed frequencies of different categories to the expected frequencies based on a theoretical distribution. In this case, you'll compare the observed number of each color in the 60 bulbs to the expected number of each color based on the Mendelian theory ratio.
Here are the steps to conduct the chi-square goodness-of-fit test using Minitab:
1. Open Minitab and enter the observed frequencies of each color into a column. Let's say you enter the observed frequencies in column C1, where C1 contains the frequencies: 27, 10, 17, 6.
2. In an empty column, calculate the expected frequencies for each color based on the Mendelian ratio. You can use the formula: expected frequency = (total number of bulbs) x (Mendelian ratio). Let's say you enter the expected frequencies in column C2.
- The total number of bulbs is 60, so the expected frequencies for each color would be:
- Blue: (60) x (6/15) = 24
- Red: (60) x (4/15) = 16
- Violet: (60) x (3/15) = 12
- Orange: (60) x (2/15) = 8
3. In an empty column, calculate the differences between the observed and expected frequencies for each color. You can subtract the value in C1 from the value in C2, and enter the differences in column C3.
4. In another empty column, calculate the squared differences between the observed and expected frequencies. You can square the value in C3, and enter the squared differences in column C4.
5. Use the chi-square goodness-of-fit test in Minitab to analyze the data. Go to the menu: Stat > Tables > Chi-Square Goodness-of-Fit.
6. In the dialog box, select the column containing the observed frequencies (C1) as the "Observations" variable. Select the column containing the expected frequencies (C2) as the "Expected frequencies" variable. Make sure the "Other options" box is checked, and enter the significance level (α) as 0.05.
7. Click "OK" to run the test.
Interpreting the results and stating the conclusion:
After running the chi-square goodness-of-fit test in Minitab, you will obtain a p-value. The p-value represents the probability of obtaining the observed data if the true Waima orchids are as expected based on the Mendelian ratio.
If the p-value is less than the chosen significance level (α = 0.05), it indicates that the observed data significantly deviate from the expected distribution. In this case, it would suggest that the bulbs purchased by the flower shop are not true Waimea orchids but a similar hybrid.
On the other hand, if the p-value is greater than the significance level, you would fail to reject the null hypothesis, indicating that the observed data does not significantly differ from the expected distribution. In this case, it would suggest that the bulbs are indeed true Waimea orchids.
Therefore, interpret the results by comparing the obtained p-value with the significance level, and state the conclusion in terms of the data.
To test the hypothesis that the bulbs purchased by the flower shop are not Waimea orchids but a similar appearing hybrid, we can use the chi-square goodness-of-fit test.
Let's conduct this test at α = 0.05 using Minitab:
1. Open Minitab and enter the observed counts for each color category in a column, let's say "Observed Count":
27
10
17
6
2. Now, enter the expected counts for each color category in another column, let's say "Expected Count":
60*(6/15)
60*(4/15)
60*(3/15)
60*(2/15)
3. Go to Stat > Tables > Chi-Square Goodness-of-Fit.
4. In the dialog box, select the "Observed Counts" column as the "Observed Frequencies" and the "Expected Count" column as the "Expected Frequencies". Make sure the "Expected Count" column is also selected as "Values."
5. Click "OK" to perform the chi-square goodness-of-fit test.
The test will calculate the chi-square statistic and the p-value.
Interpreting the results:
If the p-value is less than the chosen significance level (α), 0.05 in this case, then we reject the null hypothesis. If the p-value is greater than α, then we fail to reject the null hypothesis.
State your conclusion in terms of the data:
If the p-value is less than 0.05, we would conclude that the observed frequencies differ significantly from the expected frequencies based on the Mendelian theory, suggesting that the bulbs purchased are not true Waimea orchids but a similar hybrid.
Note: Since we cannot directly access Minitab to perform the steps, you can follow these instructions when using Minitab software for conducting the test.