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Chapter 10: Problem 48

Exercises 48 to 50 refer to the following setting. Do birds learn to time their breeding? Blue titmice eat caterpillars. The birds would like lots of caterpillars around when they have young to feed, but they must breed much earlier. Do the birds learn from one year鈥檚 experience when to time their breeding next year? Researchers randomly assigned 7 pairs of birds to have the natural caterpillar supply supplemented while feeding their young and another 6 pairs to serve as a control group relying on natural food supply. The next year, they measured how many days after the caterpillar peak the birds produced their nestlings.\(^{35}\) Did the randomization produce similar groups? First, compare the two groups of birds in the first year. The only difference should be the chance effect of the random assignment. The study report says: 鈥淚n the experimental year, the degree of synchronization did not differ between food-supplemented and control females.鈥� For this comparison, the report gives \(t=-1.05\). (a) What type of t statistic (one-sample, paired, or two-sample) is this? Justify your answer. (b) Explain how this value of t leads to the quoted conclusion.

Short Answer

Expert verified
(a) Two-sample t-test; comparing two independent groups. (b) The t-value of -1.05 indicates no significant difference between the groups.

Step by step solution

01

Understand the Scenario

In this study, researchers wanted to test if blue titmice learn to time their breeding by using a controlled experiment. They assigned some birds to a supplemented food supply group and others to a control group. The idea was to see if the breeding time in the next year was influenced by their first-year experience.
02

Identify Type of t-statistic

In the context, researchers are comparing two independent groups (food-supplemented vs. control) based on their time of breeding after the caterpillar peak. This suggests examining whether two different groups have significant differences in their means. Thus, this involves a two-sample t-test because there are two distinct independent samples being compared.
03

Analyze the t-value

The study provides a t-statistic value of \( t = -1.05 \). Given that a t-test is often used to determine if there are significant differences between the means of two groups, this specific value suggests how far the sample mean difference deviates from the null hypothesis of equal group means, in units of standard errors.
04

Conclude from t-value

A t-value of \(-1.05\) is relatively small, indicating that the means of the two groups are not significantly different at common significance levels (such as 0.05). The negative sign indicates the direction of difference but is not as crucial here. The researchers state there was no difference, implying the results were not statistically significant, supporting their conclusion stated in the report.

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Key Concepts

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

Two-sample t-test
A two-sample t-test is used when comparing the means of two independent groups to determine if there is a statistically significant difference between them. In the context of the bird experiment, the researchers wanted to see if the experience of food supplementation versus natural feeding affected the synchronization of breeding.

To perform a two-sample t-test, you calculate the means of both groups (food-supplemented and control) and assess whether these means are significantly different. The test takes into account:
  • The means of the two groups
  • The standard deviation of each group
  • The sample sizes of both groups
In this study, the resulting t-value was \(-1.05\). This value indicated that there was no significant difference in the synchronization of breeding between the two groups in the first year, leading to the conclusion that the birds did not learn to time their breeding based on the supplemented food supply.
Randomized Experiment
A randomized experiment is a type of scientific study used to establish cause and effect relationships by randomly assigning participants to different groups. This method aims to control biases and ensure that all other conditions remain equal between comparison groups.

In this study, each pair of birds was randomly assigned to either the food-supplemented group or the control group. This randomization helps to distribute any unknown factors that could affect the outcome across the groups evenly, thus enhancing the validity of the comparison.
  • Reduces selection bias
  • Creates equivalent comparison groups
  • Allows for causal conclusions to be drawn
By using a randomized design, the researchers hoped to accurately assess whether the feeding experience influenced the timing of breeding in the subsequent year.
Statistical Significance
Statistical significance is a measure that indicates whether the result of an analysis is likely due to a specific factor rather than by random chance. This concept helps researchers determine the reliability of their findings.

In the blue titmice study, the t-test resulted in a t-value of \(-1.05\). The researchers noted that this value was not significant, meaning that the difference in breeding synchronization was likely due to random variation, rather than the treatment effect of food supplementation.
  • Helps decide if differences could be due to chance
  • Common significance levels: 0.05, 0.01
  • Lack of significance indicates no strong evidence of effect
Since the t-value did not exceed common thresholds for significance, the study's conclusion was that the treatment effect was not strong enough to influence breeding timing.
Experimental Design
Experimental design refers to how an experiment is structured. It involves planning how to collect data and determine the best method to answer the research question.

For the blue titmice study, the researchers designed an experiment where 13 pairs of birds were divided into a treatment group (food-supplemented) and a control group. The purpose was to observe the breeding times relative to caterpillar presence in the next year.
  • Defines control and treatment groups
  • Ensures all variables are controlled except the independent variable
  • Provides a framework for collecting and analyzing data
This carefully structured design allowed the researchers to focus on their question: Does feeding in the first year impact the timing of breeding in the next year?

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Most popular questions from this chapter

In Exercises 39 to 42, determine whether or not the conditions for using two- sample t procedures are met. Literacy rates Do males have higher average literacy rates than females in Islamic countries? The table below shows the percent of men and women at least 15 years old who were literate in 2008 in the major Islamic nations. (We omitted countries with populations of less than 3 million.) Data for a few nations, such as Afghanistan and Iraq, were not available.\(^{30}\) $$\begin{array}{lll}{\text { Country }} & {\text { Female percent }} & {\text { Male percent }} \\ {\text { Algeria }} & {66} & {94} \\ {\text { Bangladesh }} & {48} & {71} \\ {\text { Egypt }} & {58} & {88} \\ {\text { lran }} & {77}& {97} \\ {\text { Jordan }} & {87} & {99} \\ {\text { Kazakhstan }} & {100} & {100}\\\\{\text { Lebanon }} & {86} & {98} \\ {\text { Libya }} & {78} & {100}\\\\{\text { Libya }} & {78} & {100} \\ {\text { Malaysia }} & {90} & {98}\\\ {\text { Morocoo }} & {43}& {84} \\ {\text { Saudi Arabia }} & {79} & {98}\\\\{\text { Syria }} & {77} & {95} \\ {\text { Taijkistan }} & {100} & {100} \\ {\text { Tunisia }} & {69} & {97}\\\\{\text { Turkey }} & {81}& {99} \\ {\text { Uzbekistan }} & {96} & {99} \\ {\text { Yemen }} & {41} & {93}\end{array}$$

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