A recent study ^2 examines the impact of a mother's voice on...

A recent study $^{2}$ examines the impact of a mother's voice on stress levels in young girls. The study included 68 girls ages 7 to 12 who reported good relationships with their mothers. Each girl gave a speech and then solved mental arithmetic problems in front of strangers. Cortisol levels in saliva were measured for all girls and were high, indicating that the girls felt a high level of stress from these activities. (Cortisol is a stress hormone and higher levels indicate greater stress.) After the stress-inducing activities, the girls were randomly divided into four equal-sized groups: one group talked to their mothers in person, one group talked to their mothers on the phone, one group sent and received text messages with their mothers, and one group had no contact with their mothers. Cortisol levels were measured before and after the interaction with mothers and the change in the cortisol level was recorded for each girl. (a) What are the two main variables in this study? Identify each as categorical or quantitative. (b) Is this an experiment or an observational study? (c) The researchers are testing to see if there is a difference in the change in cortisol level depending on the type of interaction with mom. What are the null and alternative hypotheses? Define any parameters used. (d) What are the total degrees of freedom? The $d f$ for groups? The $d f$ for error? (e) The results of the study show that hearing mother's voice was important in reducing stress levels. Girls who talk to their mother in person or on the phone show decreases in cortisol significantly greater, at the $5 \%$ level, than girls who text with their mothers or have no contact with their mothers. There was not a difference between in person and on the phone and there was not a difference between texting and no contact. Was the p-value of the original ANOVA test above or below $0.05 ?$

A recent study $^{2}$ examines the impact of a mother's voice on stress levels in young girls. The study included 68 girls ages 7 to 12 who reported good relationships with their mothers. Each girl gave a speech and then solved mental arithmetic problems in front of strangers. Cortisol levels in saliva were measured for all girls and were high, indicating that the girls felt a high level of stress from these activities. (Cortisol is a stress hormone and higher levels indicate greater stress.) After the stress-inducing activities, the girls were randomly divided into four equal-sized groups: one group talked to their mothers in person, one group talked to their mothers on the phone, one group sent and received text messages with their mothers, and one group had no contact with their mothers. Cortisol levels were measured before and after the interaction with mothers and the change in the cortisol level was recorded for each girl.
(a) What are the two main variables in this study? Identify each as categorical or quantitative.
(b) Is this an experiment or an observational study?
(c) The researchers are testing to see if there is a difference in the change in cortisol level depending on the type of interaction with mom. What are the null and alternative hypotheses? Define any parameters used.
(d) What are the total degrees of freedom? The $d f$ for groups? The $d f$ for error?
(e) The results of the study show that hearing mother's voice was important in reducing stress levels. Girls who talk to their mother in person or on the phone show decreases in cortisol significantly greater, at the $5 \%$ level, than girls who text with their mothers or have no contact with their mothers. There was not a difference between in person and on the phone and there was not a difference between texting and no contact. Was the p-value of the original ANOVA test above or below $0.05 ?$

Key Concepts

p-Value and Significance Level

The p-value in hypothesis testing measures the probability of obtaining results at least as extreme as the observed ones, assuming the null hypothesis is true. When the p-value is less than or equal to the predetermined significance level (commonly 0.05), the null hypothesis is rejected, suggesting there are statistically significant differences. Understanding the p-value and its relation to the significance level is key to interpreting the outcomes of the statistical test.

Hypothesis Testing in ANOVA

In ANOVA, the null hypothesis (H0) typically states that there are no differences in the population means of the response variable across the different groups, while the alternative hypothesis (Ha) asserts that at least one group mean is significantly different from the others. The parameters in these hypotheses denote the true mean changes in the outcome for each group, and testing these hypotheses helps determine whether group differences exist.

Degrees of Freedom in ANOVA

Degrees of freedom (df) are used to quantify the number of independent pieces of information available for estimating parameters. In the context of ANOVA, the total degrees of freedom is calculated as the total number of observations minus one, the between-group degrees of freedom is the number of groups minus one, and the within-group (error) degrees of freedom is the total degrees of freedom minus the between-group degrees of freedom. These values are crucial for constructing the F-statistic used in the ANOVA test.

Variable Classification

This concept involves distinguishing between categorical variables, which represent different groups or categories (such as types of interaction with mothers), and quantitative variables, which are measured on a numerical scale (such as cortisol level changes). Understanding the type of data collected is essential for determining the appropriate statistical analysis and testing methods.

Experimental vs Observational Study Design

This concept refers to the fundamental difference between experiments, where subjects are randomly assigned to treatment groups to infer causality, and observational studies, where no random assignment is made, and subjects are merely observed. Recognizing this distinction is vital in determining the strength of causal inferences that can be drawn from the study.

Transcript

00:01 In this question, we have a study of 68 girls that looked at the effect of a mother's voice on their daughters.

00:06 Each girl gave a speech and then solved math problems before strangers, which is very stressful.

00:12 Afterwards, the girls were divided into four groups, 17 each.

00:16 Group one talked to their mom in person.

00:19 Group two talked to mom on the phone.

00:21 Group three texted their moms, and group four had no contact with them.

00:25 Then the girls ' levels of cortisol, which is a stress hormone, were measured.

00:30 They were measured after they were also measured before.

00:34 So what we want to answer first is the two variables and their types.

00:39 So one variable is going to be the type of contact with mom.

00:45 So was that speaking in person or on the phone? was it texting or was there no contact? and that is a categorical variable.

00:56 The other variable, and this will be our response variable, is the change in level of cortisol.

01:03 So how about if i write cortisol level change? did the stress hormone decrease? that's what they were especially looking at after they had contact with mom.

01:15 So that's something that is a numerical variable because that level can be measured with values.

01:22 Is this an experiment or observational study? this is an experiment.

01:27 And it's an experiment because there's a controlled setting.

01:31 There were 68 girls.

01:32 They were randomly divided into four groups of 17 each, and each group was given a different treatment, and we had the results measured.

01:43 So that fits the definition of an experiment.

01:46 Now let's write the null and alternative hypotheses and describe the parameters in question here.

01:53 So the null hypothesis is that the mean cortisol change is equal for all four groups...

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