Question

The coloration of male guppies may affect the mating preference of the female guppy. To test this hypothesis, scientists first identified two types of guppies, Yarra and Paria, that display different colorations ("Evolutionary Mismatch of Mating Preferences and Male Colour Patterns in Guppies," Animal Behaviour \([1997]: 343-51\) ). The relative area of orange was calculated for fish of each type. A random sample of 30 Yarra guppies resulted in a mean relative area of \(.106\) and a standard deviation of .055. A random sample of 30 Paria guppies resulted in a mean relative area of 178 and a standard deviation \(.058\). Is there evidence of a difference in coloration? Test the relevant hypotheses to determine whether the mean area of orange is different for the two types of guppies.

Short answer

The decision of the hypothesis test will depend on the comparison between the calculated test statistic (t) and the critical value from the t-distribution table. The null hypothesis will be rejected if |t|>t-crit, indicating there is evidence of a difference in coloration between Yarra and Paria guppies. If |t|<=t-crit, then the null hypothesis is not rejected, suggesting no significant difference in coloration between the two types of guppies.

Step-by-step solution

State the Null and Alternative Hypotheses

The null hypothesis (H0) is that the mean relative area of orange is the same for both types of guppies. The alternative hypothesis (Ha) is that the mean relative area of orange is not the same for the two types. Mathematically, these can be written as H0: μ1 = μ2 and Ha: μ1 ≠ μ2, where μ1 and μ2 are the population means of Yarra and Paria guppies respectively.

Calculate the Test Statistic

The formula for the test statistic for testing the difference in means is:\[ t = \frac{\(x̄1 - x̄2\) - (μ1 - μ2)}{\sqrt{\(s1^2/n1 + s2^2/n2\)}}Where:x̄1 = .106 and x̄2 = .178 are the sample means, μ1 - μ2 = 0 (as per the null hypothesis), s1 = .055 and s2 = .058 are the standard deviations, n1 = n2 = 30 are the sample sizes.Substituting these values into the formula will give the value of t.

Find the Critical Value and Make the Decision

The critical values (t-crit) for a two-tailed test at a 5% significance level and degree of freedom (df) = n1 + n2 - 2 = 30 + 30 - 2 = 58, can be looked up from the t-distribution table. If the absolute value of t is greater than t-crit, we reject the null hypothesis. If not, we fail to reject the null hypothesis.

Key concepts

Null and Alternative Hypotheses

In the realm of hypothesis testing, two competing hypotheses form the crux of the investigation. The null hypothesis, symbolized as H0, represents the status quo or the assumption that no difference or effect exists. In our guppy coloration study, H0 posits that the mean relative areas of orange on Yarra and Paria guppies are identical.
The alternative hypothesis, denoted as Ha or H1, challenges the null by suggesting that a difference or effect does exist. For the guppies, Ha posits that Yarra and Paria guppies have different mean relative areas of orange. The investigation proceeds by assuming the null hypothesis is true and then looking for evidence to either reject it or fail to reject it. It's vital to remember that 'failing to reject' the null does not prove it true, but rather implies insufficient evidence to accept the alternative.

Test Statistic Calculation

Once hypotheses are stated, we move to calculating a test statistic. This is the compass that navigates us through the data's randomness to ascertain if the observations are more extreme than what we'd expect under the null hypothesis.
For the guppy study, the test statistic helps determine if the observed difference in mean coloration is statistically significant or a result of random chance. The formula encapsulates the difference between the sample means, standard deviation, and sample size. A calculated t-value is extracted, which then gets compared to a critical value from the t-distribution. If our test statistic exceeds the critical t-value in absolute terms, it points towards the alternative hypothesis being more plausible.

T-Distribution

The t-distribution is a blueprint for inference when dealing with small or medium-sized samples, especially when the population standard deviation is unknown. It resembles a normal distribution but has thicker tails, meaning that it predicts more variability and is forgiving towards outliers.
In hypothesis testing, the t-distribution table is like a map that demonstrates where our test statistic lies in the landscape of chance results. You can locate the critical values based on your sample's degrees of freedom, which adjusts for sample size. The larger the sample, the more the t-distribution resembles the normal distribution.

Statistical Significance

Statistical significance acts as the judge in hypothesis testing, deciding whether the observed data are convincingly unusual under the null hypothesis. Significance is often determined by a p-value, which assesses the probability of obtaining test results at least as extreme as the ones observed during the experiment, assuming that the null hypothesis is true.
Conventions in many fields establish a p-value threshold (alpha level), commonly set at 0.05. Crossing this line by obtaining a smaller p-value or a larger test statistic relative to the critical value from the t-distribution, warrants enough evidence to reject the null hypothesis. In our guppy example, if the test statistic falls into that extreme 5% end of the t-distribution, the data suggests a significant difference in coloration, leading scientists to consider the mating preference hypothesis as viable.