Question

In a field study of mating behavior in the Mormon cricket (Anabrus simplex), a biologist noted that some females mated successfully while others were rejected by the males before coupling was complete. The question arose whether some aspect of body size might play a role in mating success. The accompanying table summarizes measurements of head width \((\mathrm{mm})\) in the two groups of females. \(^{45}\) (a) Construct a \(95 \%\) confidence interval for the difference in population means. [Note: Formula (6.7.1) yields 35.7 degrees of freedom for these data. (b) Interpret the confidence interval from part (a) in the context of this setting. (c) Using your interval computed in (a) to support your answer, is there strong evidence that the population mean head width is indeed larger for successful maters than unsuccessful maters? $$ \begin{array}{|lcc|} \hline & \text { Successful } & \text { Unsuccessful } \\ \hline n & 22 & 17 \\ \bar{y} & 8.498 & 8.440 \\ s & 0.283 & 0.262 \\ \hline \end{array} $$

Short answer

The 95% confidence interval for the difference in population means of head width will indicate if there is a statistically significant difference between successful and unsuccessful maters. If the confidence interval does not include zero, there is evidence to suggest that successful maters have different mean head widths than unsuccessful maters.

Step-by-step solution

Calculate the Standard Error of Difference between Two Means

The standard error (SE) of the difference between two means can be calculated using the formula: \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \] where \( s_1 \) and \( s_2 \) are the sample standard deviations, and \( n_1 \) and \( n_2 \) are the sample sizes. For the successful group, \( s_1 = 0.283 \), \( n_1 = 22 \), and for the unsuccessful group, \( s_2 = 0.262 \), \( n_2 = 17 \). Plug these values into the formula to get the SE for the difference in head width.

calculate the 95% Confidence Interval for the Difference in Means

The 95% confidence interval is given by \( \bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2} \times SE \), where \( t_{\alpha/2} \) is the t-score for the desired confidence level and degrees of freedom (35.7 as noted in the exercise). Using the t-distribution table or calculator, find the t-score for a 95% confidence interval with approximately 36 d.f. Then, use the sample mean head widths \( \bar{y}_1 = 8.498 \) and \( \bar{y}_2 = 8.440 \) to compute the confidence interval.

Interpret the Confidence Interval

Once the confidence interval is calculated, interpret it in the context of the study. The interval gives a range of values for the difference in population means for head width between successful and unsuccessful maters. If the interval does not contain zero, it suggests a difference between the groups. In this context, it would suggest that body size, as measured by head width, is related to mating success in Mormon crickets.

Analyze the Confidence Interval for Evidence

Examine the confidence interval's bounds to determine if they both are positive or both are negative. If the entire interval lies above zero, this can be taken as strong evidence that the population mean head width for successful maters is larger than for unsuccessful maters. If the interval crosses zero, such definitive evidence does not exist.

Key concepts

Biostatistics

Biostatistics is an essential branch of statistics that applies statistical methods to biological and health-related fields. It's fundamental in the design and analysis of experiments and surveys, interpretation of data, and making informed decisions in health policies and medical research.

In our cricket mating study, biostatistics is leveraged to determine whether the variation in body size, indicated by head width, correlates with mating success. By analyzing sample data from successful and unsuccessful maters, we use statistical tools to infer about the entire cricket population, providing insights into potential biological factors affecting mating behavior.

T-Distribution

The t-distribution, particularly useful when dealing with small sample sizes, is a type of probability distribution that arises while estimating population parameters when the sample size is limited and the population standard deviation is unknown. It resembles the normal distribution but has heavier tails.

For our cricket data, with sample sizes of 22 and 17, the t-distribution is more applicable than the normal distribution for constructing confidence intervals. Since we are estimating the population mean difference of head width, and the exact population standard deviation is not known, the t-distribution provides a more adequate way to account for these uncertainties, especially with relatively few degrees of freedom, reported as 35.7 in the exercise.

Standard Error

The standard error (SE) measures the dispersion or accuracy with which a sample statistic represents a population parameter. The SE of the difference between two sample means is critical in hypothesis testing and constructing confidence intervals, as it indicates how much the sample means differ on average from the true population means.

In the context of our insect study, we calculate the SE for the difference in head width means. The lower the SE, the more precise our estimate of the population mean difference is considered to be. It’s calculated using standard deviation and sample size from both groups, highlighting the importance of these measurements in estimating variability.

Sample Size

Sample size refers to the number of individual observations used to calculate a statistic. In statistical analysis, a larger sample size generally leads to a smaller standard error, which means more precise estimates of the population parameter.

Our cricket study has sample sizes of 22 for the successful maters and 17 for the unsuccessful group. The choice of sample size affects the standard error and consequently the width of the confidence interval. In this case, the choice of relatively small sample sizes is typical in field studies but it demands careful statistical considerations, such as the use of t-distribution, to draw reliable conclusions about the population.

Population Mean Difference

The population mean difference is a parameter that quantifies the difference in average between two groups within a population. It's often the focus of comparative studies like ours, which are aimed at identifying whether such a difference exists.

In the context of the Mormon cricket study, we are interested in the difference in population means of head width between females that had mating success and those that did not. The confidence interval provides a range where this true difference is estimated to lie with a certain level of confidence, in our case, 95%. If the confidence interval does not include zero, it suggests a statistically significant difference in mean head widths, pointing towards the significance of body size in mating success.