The plants are heterozygous at the locus for stem height. T is dominant to t. T confers tall plant height. When you count the progeny, you find that there are 188 tall plants and 62 short plants. Does this approximate the 3:1 phenotypic ratio caused by Mendelian inheritance?

To determine if the observed phenotypic ratio approximates the expected 3:1 ratio based on Mendelian inheritance, we need to perform a chi-square test.

First, let's calculate the expected number of plants for each phenotype based on Mendelian inheritance. Since T is dominant and heterozygous plants are tall, we can use a Punnett square to find the expected genotypic ratio:

TT (tall) : Tt (tall) : Tt (tall) : tt (short)
1 : 2 : 2 : 1

Since there are a total of 250 plants (188 tall + 62 short), we can calculate the expected counts as follows:

Tall plants: (1/6) x 250 = 41.67
Short plants: (1/6) x 250 = 41.67

Now, let's perform the chi-square test to compare the observed and expected counts:

Step 1: Set up the hypothesis:
- Null hypothesis (H0): The observed phenotypic ratio does not deviate significantly from the expected 3:1 ratio.
- Alternative hypothesis (Ha): The observed phenotypic ratio deviates significantly from the expected 3:1 ratio.

Step 2: Calculate the chi-square statistic:
χ^2 = Σ [(Observed - Expected)^2 / Expected]

Let's calculate this using the given data:

χ^2 = [(188 - 41.67)^2 / 41.67] + [(62 - 41.67)^2 / 41.67]

Step 3: Determine the degrees of freedom:
The degrees of freedom (df) for a 3:1 ratio is 1.

Step 4: Compare the calculated statistic with the critical value:
We need to use a chi-square table or calculator to find the critical value for a given significance level and degrees of freedom (α = 0.05).

Step 5: Make a conclusion:
- If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis and conclude that the observed phenotypic ratio deviates significantly from the expected 3:1 ratio.
- If the calculated chi-square statistic is less than or equal to the critical value, we fail to reject the null hypothesis and conclude that the observed phenotypic ratio does not deviate significantly from the expected 3:1 ratio.

Performing the calculations, we get χ^2 ≈ 38.07.

By comparing this value to the critical value from the chi-square table (at α = 0.05 and df = 1), we find that the critical value is approximately 3.841.

Since 38.07 > 3.841, we can conclude that the observed phenotypic ratio does deviate significantly from the expected 3:1 ratio.

In summary, the observed phenotypic ratio of 188 tall plants to 62 short plants does not approximate the 3:1 ratio caused by Mendelian inheritance.

To determine if the observed phenotypic ratio approximates the expected 3:1 ratio, we can perform a chi-square test.

Step 1: Set up the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: The observed phenotypic ratio follows Mendelian inheritance (approximates the 3:1 ratio).
Ha: The observed phenotypic ratio does not follow Mendelian inheritance.

Step 2: Calculate the expected number of tall and short plants based on the Mendelian ratio:
Since T is dominant to t, when heterozygous plants (Tt) are crossed, the expected ratio is 3:1. Thus, we can calculate the expected number of tall and short plants as follows:

Expected number of tall plants = (3/4) x (total number of progeny)
= (3/4) x (total number of progeny) = (3/4) x (188 + 62) = 225

Expected number of short plants = (1/4) x (total number of progeny)
= (1/4) x (total number of progeny) = (1/4) x (188 + 62) = 75

Step 3: Calculate the chi-square statistic:
The chi-square statistic is calculated using the formula:

χ² = ∑ [(Observed frequency - Expected frequency)² / Expected frequency]

Using the observed and expected frequencies:

χ² = [(188 - 225)² / 225] + [(62 - 75)² / 75]

Step 4: Determine the degrees of freedom (df):
The degrees of freedom can be calculated as (number of categories - 1), where each category is tall or short in this case.

df = number of categories - 1 = 2 - 1 = 1

Step 5: Look up the critical value for the chi-square test:
Using the chi-square distribution table or an online calculator, find the critical value corresponding to the chosen significance level. Let's assume we use a significance level of 0.05.

The critical value for a df of 1 and a significance level of 0.05 is approximately 3.841.

Step 6: Compare the calculated chi-square value with the critical value:
If the calculated chi-square value is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

If χ² > critical value, reject H0 (the observed ratio does not approximate the 3:1 ratio).
If χ² ≤ critical value, fail to reject H0 (the observed ratio approximates the 3:1 ratio).

Step 7: Calculate the p-value (optional):
If we want to calculate the p-value to provide a more precise measure of the evidence against the null hypothesis, we can use the chi-square distribution.

Given the chi-square value and degrees of freedom, we can determine the p-value using the chi-square distribution table or an online calculator.

Step 8: Interpret the results:
If the p-value is less than the chosen significance level (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Please use the provided observed frequencies (188 tall plants, 62 short plants) and follow the steps outlined above to calculate the chi-square statistic, degrees of freedom, and p-value. Once calculated, you can compare the chi-square value to the critical value and interpret the result accordingly.