91Ó°ÊÓ

The following statistics are the result of an experiment conducted by P. I. Ward to investigate a theory concerning the molting behavior of the male Gammarus pulex, a small crustacean. \(^{\star}\) If a male needs to molt while paired with a female, he must release her, and so loses her. The theory is that the male \(G\). pulex is able to postpone molting, thereby reducing the possibility of losing his mate. Ward randomly assigned 100 pairs of males and females to two groups of 50 each. Pairs in the first group were maintained together (normal); those in the second group were separated (split). The length of time to molt was recorded for both males and females, and the means, standard deviations, and sample sizes are shown in the accompanying table. (The number of crustaceans in each of the four samples is less than 50 because some in each group did not survive until molting time.) a. Find a \(99 \%\) confidence interval for the difference in mean molt time for "normal" males versus those "split" from their mates. b. Interpret the interval.

Step by step solution

01

Identify the Statistical Information

From the problem, we have data for two groups: 'normal' males (males with female) and 'split' males (males without female). We are given the means, standard deviations, and sample sizes for these groups. Let's assume: Mean A = \( \bar{x}_1 \), SD A = \( s_1 \), sample size A = \( n_1 \) for 'normal' males; and Mean B = \( \bar{x}_2 \), SD B = \( s_2 \), sample size B = \( n_2 \) for 'split' males.

02

Formula for Confidence Interval

The formula to calculate the confidence interval for the difference between two means is given by: \[ (\bar{x}_1 - \bar{x}_2) \pm z \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]Where \( z \) is the z-score corresponding to the desired confidence level, which is \(99\%\) in our case.

03

Find the Z-score for 99% Confidence

For a 99% confidence interval, the z-score (critical value) can be found from a standard normal distribution table. The z-score corresponding to a 99% confidence level is approximately 2.576.

04

Calculate the Standard Error

Use the given standard deviations \( s_1 \) and \( s_2 \), and the sample sizes \( n_1 \) and \( n_2 \) to calculate the standard error:\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \]

05

Calculate the Confidence Interval

Substitute \( \bar{x}_1 \), \( \bar{x}_2 \), the z-score value, and the calculated standard error into the confidence interval formula:\[ (\bar{x}_1 - \bar{x}_2) \pm 2.576 \times SE \]

06

Interpret the Interval

Once the confidence interval is computed, it gives the range within which the true difference in mean molt times for 'normal' versus 'split' males lies with 99% confidence. If this interval does not include zero, it suggests a significant difference in molt times between the two groups, affirming or refuting the crustaceans' ability to postpone molting.

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Key Concepts

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

Understanding the Difference of Means

When comparing two distinct groups, it's essential to understand the concept of "difference of means." In this context, it involves comparing the average molt times of 'normal' males (those with a female) to 'split' males (those without a female). The difference between these two means helps us assess whether the group condition (being with or without a mate) affects the time to molt.

The formula for calculating the difference of means is simple: subtract one group's mean from the other:

• Difference of Means = Mean of Group A - Mean of Group B

This difference is a crucial part of inference to determine whether any observed difference is statistically significant or likely due to random variation.

By creating a confidence interval around this difference, researchers can assess whether the difference is representative of the entire population of crustaceans.

Basics of Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to determine if there are significant differences between groups, like our 'normal' and 'split' males. A hypothesis test involves a null hypothesis, typically stating that there is no difference between the groups, and an alternative hypothesis, suggesting a difference exists.

Here's a simple outline of hypothesis testing:

• Null Hypothesis (H0): There is no difference in mean molt times between 'normal' and 'split' males.
• Alternative Hypothesis (H1): There is a difference in mean molt times.
• Determine the significance level (often 0.05 or 0.01).
• Use statistical tools to analyze the data and calculate a p-value or construct a confidence interval.

If the calculated confidence interval for the difference of means does not include zero, it suggests that the difference observed is statistically significant, thus rejecting the null hypothesis.

The Z-Score in Confidence Intervals

The z-score plays a critical role in constructing confidence intervals, especially when the population standard deviations are known or the sample size is large. It reflects how many standard deviations away a data point is from the mean in a standard normal distribution.

For our problem, a 99% confidence interval is required. The z-score for this confidence level is approximately 2.576. This value is derived from standard normal distribution tables and indicates that the constructed interval will capture the true difference of means 99% of the time.

Including the z-score in the confidence interval formula helps set the range within which we believe the true difference lies. It accounts for variability in the data and helps estimate the precision and reliability of our observed difference.

Calculating the Standard Error

The standard error is a measure of the variability or dispersion of the sample mean from the true population mean. In the context of comparing two means, it gives insight into how much the means of the samples are expected to fluctuate.

For our exercise, the standard error (SE) is calculated using the formula:

• SE = \( \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \)

Here, \( s_1 \) and \( s_2 \) represent the standard deviations of the 'normal' and 'split' male groups, while \( n_1 \) and \( n_2 \) represent the sample sizes.

Calculating the standard error is a vital step in determining the confidence interval. It accommodates the sample sizes and variability of the data, providing a clearer picture of how confident we can be in our results. A smaller standard error indicates that the sample means are close to the true difference, enhancing the reliability of our confidence interval.