In Exercises 13–20, determine whether the results appear to have statistical significance, and also determine whether the results appear to have practical significance.

Mendel’s Genetics Experiments One of Gregor Mendel’s famous hybridization experiments with peas yielded 580 offspring with 152 of those peas (or 26%) having yellow pods. According to Mendel’s theory, 25% of the offspring peas should have yellow pods.

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Step 1: Define the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis is that the proportion of peas with yellow pods is 25% (p = 0.25), as predicted by Mendel's theory. The alternative hypothesis is that the proportion is different from 25% (p ≠ 0.25).

Step 2: Calculate the sample proportion (p̂) of peas with yellow pods. This is done by dividing the number of peas with yellow pods (152) by the total number of peas (580).

Step 3: Determine the standard error of the sample proportion. The standard error (SE) is calculated using the formula: SE = sqrt((p * (1 - p)) / n), where p is the hypothesized population proportion (0.25) and n is the sample size (580).

Step 4: Calculate the test statistic (z-score) using the formula: z = (p̂ - p) / SE, where p̂ is the sample proportion, p is the hypothesized population proportion, and SE is the standard error calculated in the previous step.

Step 5: Compare the calculated z-score to the critical z-value for a chosen significance level (commonly 0.05 for a two-tailed test). If the absolute value of the z-score is greater than the critical z-value, the null hypothesis is rejected, indicating statistical significance. Additionally, consider whether the difference between the observed and expected proportions is large enough to be of practical significance.

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주요 개념

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Statistical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. In the context of Mendel's experiment, it involves determining whether the observed 26% of yellow pods significantly deviates from the expected 25% under Mendel's theory, using a hypothesis test and a significance level (e.g., 0.05).

Practical Significance

Practical significance considers whether the size of the effect or difference is large enough to be meaningful in a real-world context. Even if the 1% difference in Mendel's experiment is statistically significant, it may not be practically significant if it doesn't have a substantial impact on the understanding or application of genetic principles.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis. In Mendel's experiment, the null hypothesis might state that the proportion of yellow pods is 25%, and a test (e.g., chi-square test) would determine if the observed 26% provides sufficient evidence to reject this hypothesis.