Question

In a study of resistance to a certain soybean virus, biologists cross- fertilized two soybean cultivars. They expected to get a 3: 1 ratio of resistant to susceptible plants. The observed data were 58 resistant and 26 suscep- tible plants. \(^{50}\) Are these data significantly inconsistent with the expected 3: 1 ratio? Consider a test, using \(\alpha=0.10\) and a nondirectional alternative. (a) What are the expected counts for the two categories under the null hypothesis? (b) The \(P\) -value for the chi-square test (with a nondirectional alternative) is \(0.21 .\) If \(\alpha=0.10,\) what is your conclusion regarding \(H_{0} ?\) (c) Do these results confirm the 3: 1 ratio expected by the researchers?

Short answer

The expected counts are 63 for resistant and 21 for susceptible plants. Since the P-value (0.21) is greater than the significance level (0.10), we do not reject the null hypothesis; thus, the observed data are not significantly inconsistent with the expected 3:1 ratio.

Step-by-step solution

Calculate Total Number of Plants

Add the number of resistant plants and susceptible plants together to get the total number of plants.

Calculate Expected Counts

Determine the expected counts for both resistant and susceptible plants using the 3:1 ratio with total number of plants.

Evaluate the P-value Against the Significance Level

Compare the P-value from the chi-square test to the significance level \(\alpha=0.10\)).

State the Conclusion

Make a decision on the null hypothesis based on the comparison between the P-value and the significance level.

Assess Confirmation of Ratio

Use the conclusion from the chi-square test to assess if it confirms the expected 3:1 ratio by the researchers.

Key concepts

Null Hypothesis

Imagine you’re trying to decode a secret message. The null hypothesis is like starting with the assumption that there is no hidden message, just random letters. In scientific research, the null hypothesis (\( H_0 \) ) claims that there is no effect or no relationship between two measured phenomena. For our soybean virus study, the null hypothesis states that the ratio of resistant to susceptible plants will be exactly 3:1, as per genetic expectations.

To see if this secret message might really exist, scientists use a test. If the test shows that it's very unlikely to get the given data if the null hypothesis were true, researchers might start thinking that there could indeed be a hidden message after all - that is, the alternative hypothesis might be correct. In the soybean study, this translates to checking if the observed number of resistant and susceptible plants is highly unlikely if the true ratio was genuinely 3:1.

P-value

The P-value helps you interpret the test results - it's like a lie detector for the null hypothesis. Specifically, it's the probability of observing data as extreme as what was actually observed, assuming the null hypothesis is true. It's not the probability that the null hypothesis is correct (a common misconception). A high P-value suggests that the observed data are fairly likely under the null hypothesis—meaning, no reason to suspect a lie. A low P-value indicates that what we're observing is pretty rare if the null hypothesis were true, making us doubt the null hypothesis.

In our case, the P-value is 0.21. This would be like the lie detector showing that there's only a 21% chance that the patterns in the observed data would occur if the ratio really was 3:1—kind of a 'maybe' in terms of if there’s a hidden message, not strong enough to make us truly suspicious.

Statistical Significance

Let's equate statistical significance to a threshold in a high-jump competition. Just like how the bar is set at a certain height, researchers choose a threshold called the significance level (\( \text{alpha, or } \) ) when they're testing the null hypothesis. Jumping above the bar means an athlete has cleared the height; similarly, a P-value below the significance level indicates that the observed data are unlikely enough under the null hypothesis that we reject it.

Our significance level is 0.10, or 10%. Since the P-value (0.21) in our soybean study is higher than 0.10, we don't have enough evidence to reject the null hypothesis. It's like an athlete who can't jump high enough to clear the bar; we can't conclusively say there's a significant difference from what was expected.

Genetic Resistance

In nature's lottery, genetic resistance is a winning ticket that some organisms have against particular threats - in this case, a soybean virus. It's a trait that can be passed on to offspring, offering a shield against disease. Think of it as a genetic armor that certain plants have, making them resistant to specific viruses. In our textbook problem, some soybeans have this armor against the virus, and some don't.

When scientists cross-fertilize two different cultivars, they're essentially trying to figure out the pattern of inheritance for this genetic armor. Will it be distributed so that three out of four plants can resist the virus, or not? That's what the 3:1 ratio is all about: predicting the inheritance of resistance in the progeny.

Soybean Virus Study

The context for this entire problem is a soybean virus study conducted by biologists. They're not just randomly guessing what the ratio of resistant to susceptible plants should be; it's based on established genetic principles. When they cross-fertilize two types of soybeans, they expect to see the traits of genetic resistance passed down in a certain way — that's Mendel's laws of inheritance in action!

But much like detective work, scientists have to look at evidence to confirm their hypotheses. They count how many plants are resistant and how many are not, then use this count to perform a statistical test - in this case, the chi-square test. The test helps them determine if the observed data match their expectations or if something surprising is happening, which could indicate a complexity in the genetic resistance or a fluke in the data collection.