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Chapter 9: Problem 27

An experiment to determine the effects of temperature on the survival of insect eggs was described in the article "Development Rates and a Temperature- Dependent Model of Pales Weevil" (Environ. Entomology, 1987:956-962). At \(11^{\circ} \mathrm{C}, 73\) of 91 eggs survived to the next stage of development. At \(30^{\circ} \mathrm{C}, 102\) of 110 eggs survived. Do the results of this experiment suggest that the survival rate (proportion surviving) differs for the two temperatures? Calculate the \(P\)-value and use it to test the appropriate hypotheses.

Short Answer

Expert verified
Significant difference in survival rates; P-value = 0.0078.

Step by step solution

01

Define Hypotheses

We start by defining the null and alternative hypotheses. Let \( p_1 \) be the proportion of eggs surviving at \( 11^{\circ} \mathrm{C} \) and \( p_2 \) be the proportion of eggs surviving at \( 30^{\circ} \mathrm{C} \). The null hypothesis is \( H_0: p_1 = p_2 \), meaning there is no difference in survival rates. The alternative hypothesis is \( H_a: p_1 eq p_2 \), indicating there is a difference in survival rates.
02

Calculate Sample Proportions

Calculate the sample proportions for both temperature groups. For \( 11^{\circ} \mathrm{C} \), \( \hat{p}_1 = \frac{73}{91} \approx 0.8022 \). For \( 30^{\circ} \mathrm{C} \), \( \hat{p}_2 = \frac{102}{110} \approx 0.9273 \).
03

Find the Pooled Proportion

To find the pooled proportion \( \hat{p} \), use the formula \( \hat{p} = \frac{x_1 + x_2}{n_1 + n_2} \) where \( x_1 = 73 \), \( x_2 = 102 \), \( n_1 = 91 \), \( n_2 = 110 \). Thus, \( \hat{p} = \frac{73 + 102}{91 + 110} = \frac{175}{201} \approx 0.8706 \).
04

Calculate Standard Error

Calculate the standard error using the formula \( \text{SE} = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \). Substituting in our values: \( \text{SE} = \sqrt{0.8706 \times 0.1294 \left( \frac{1}{91} + \frac{1}{110} \right)} \approx 0.047 \).
05

Compute Z-statistic

The Z-statistic is calculated using \( Z = \frac{\hat{p}_1 - \hat{p}_2}{\text{SE}} \). Substituting our values: \( Z = \frac{0.8022 - 0.9273}{0.047} \approx -2.6596 \).
06

Determine the P-value

Use the standard normal distribution to find the P-value associated with the Z-statistic. A Z-value of approximately -2.6596 corresponds to a P-value of about 0.0078 (two-tailed).
07

Make a Conclusion

As the P-value (0.0078) is less than 0.05, we reject the null hypothesis. There is sufficient evidence to suggest a difference in survival rates between the two temperature conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion
When we talk about proportion in statistical hypothesis testing, we're referring to the fraction of a total number of cases that show a particular characteristic. In this experiment, the survival rate of insect eggs at different temperatures represents the focus of our proportion analysis.
A proportion is represented as a ratio:
  • For \(11^{\circ} \mathrm{C}\), the proportion of surviving eggs is calculated as \( \hat{p}_1 = \frac{73}{91} \approx 0.8022 \) which means approximately 80.22% of the eggs survived.
  • For \(30^{\circ} \mathrm{C}\), it鈥檚 \( \hat{p}_2 = \frac{102}{110} \approx 0.9273 \), representing around 92.73% survival.
In statistical terms, we compare these proportions to determine whether the survival rates differ significantly at these two different temperatures.
Z-statistic
The Z-statistic is a measurement that describes the number of standard deviations a data point is from the mean of a distribution. It's vital for testing hypotheses about gene-level differences between groups, like in this problem, based on survival rates.
Calculating the Z-statistic is a pivotal step:
  • We take the difference in sample proportions: \( \hat{p}_1 - \hat{p}_2 \)
  • This difference is then divided by the standard error:
  • For our experiment, the Z-statistic was calculated as \( Z = \frac{0.8022 - 0.9273}{0.047} \approx -2.6596 \).
A Z-statistic helps us understand how different the sample proportions are in terms of standard deviations. This is essential in hypothesis testing, as it assists in judging whether the differences in proportions are significant.
P-value
In hypothesis testing, the P-value helps to determine the significance of results obtained in an experiment. It quantifies the probability that the observed data would occur by random chance under the null hypothesis.
Here's what you need to know about the P-value:
  • It tells us the likelihood of observing a test statistic as extreme or more extreme than the one calculated.
  • With a Z-statistic of approximately -2.6596, the corresponding P-value arrived at is about 0.0078.
  • If the P-value is lower than the significance level (often set at 0.05), the null hypothesis is rejected.
In this case, since 0.0078 is less than 0.05, it suggests that there's a statistically significant difference in the survival proportions of eggs at different temperatures.
Standard Error
Standard error (SE) quantifies the amount of variation or "spread" in a sampling distribution. It's a key part of determining the reliability of a proportion gap in hypothesis tests.
The standard error is especially crucial when calculating the Z-statistic:
  • The formula for SE when comparing two proportions is: \[ SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \]
  • Here, \( \hat{p} \) is the pooled proportion, combining observations from both groups. For this problem, \( \hat{p} = \frac{175}{201} \approx 0.8706 \).
  • Substituting the appropriate numbers, the SE for this experiment was calculated to be approximately 0.047.
The standard error provides an estimation of the difference's precision between the group proportions, ensuring that any analysis on statistical significance considers the observed variability.

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Most popular questions from this chapter

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