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Psychology: Self-Esteem Female undergraduates in randomized groups of 15 took part in a self-esteem study ("There's More to Self-Esteem than Whether It Is High or Low: The Importance of Stability of Self-Esteem," by M. H. Kernis et al., Journal of Personality and Social Psychology, Vol. 65, No. 6). The study measured an index of self-esteem from the points of view competence, social acceptance, and physical attractiveness. Let \(x_{1}, x_{2}\), and \(x_{3}\) be random variables representing the measure of self-esteem through \(x_{1}\) (competence), \(x_{2}\) (social acceptance), and \(x_{3}\) (attractiveness). Higher index values mean a more positive influence on self-esteem. $$ \begin{array}{ccccc} \hline \text { Variable } & \text { Sample Size } & \text { Mean } \bar{x} & \text { Standard Deviation } s & \text { Population Mean } \\ \hline x_{1} & 15 & 19.84 & 3.07 & \mu_{1} \\ x_{2} & 15 & 19.32 & 3.62 & \mu_{2} \\ x_{3} & 15 & 17.88 & 3.74 & \mu_{3} \\ \hline \end{array} $$ (a) Find an \(85 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). (b) Find an \(85 \%\) confidence interval for \(\mu_{1}-\mu_{3} .\) (c) Find an \(85 \%\) confidence interval for \(\mu_{2}-\mu_{3}\). (d) Interpretation Comment on the meaning of each of the confidence intervals found in parts (a), (b), and (c). At the \(85 \%\) confidence level, what can you say about the average differences in influence on self-esteem between competence and social acceptance? between competence and attractiveness? between social acceptance and attractiveness?

Step by step solution

01

Identify the Confidence Interval Formula

To find the confidence intervals, we'll use the formula for the difference between two means: \[CI = (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n} + \frac{s_2^2}{n}}\]where \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means, \( s_1 \) and \( s_2 \) are the standard deviations, and \( n \) is the sample size. \( t_{\alpha/2} \) is the critical t-value for \( 85\% \) confidence and \( 14 \) degrees of freedom.

02

Calculate Critical t-value

For a \( 85\% \) confidence interval and \( 14 \) degrees of freedom (since sample size \( n = 15 \)), the critical t-value \( t_{\alpha/2} \) can be found using a t-distribution table or calculator. \( t_{0.075}(14) = 1.341 \, (approx) \).

03

Calculate CI for 渭鈧� - 渭鈧�

Using the values: \( \bar{x}_1 = 19.84 \), \( \bar{x}_2 = 19.32 \), \( s_1 = 3.07 \), \( s_2 = 3.62 \), and \( n = 15 \), substitute into the equation:\[CI = (19.84 - 19.32) \pm 1.341 \cdot \sqrt{\frac{3.07^2}{15} + \frac{3.62^2}{15}}\]Calculating further: \[CI = 0.52 \pm 1.341 \cdot \sqrt{\frac{9.4249}{15} + \frac{13.1044}{15}}\]\[CI = 0.52 \pm 1.341 \cdot 1.354\]\[CI = 0.52 \pm 1.815\]\[CI = (-1.295, 2.335)\]

04

Calculate CI for 渭鈧� - 渭鈧�

Repeat for \( \mu_1 - \mu_3 \): Using \( \bar{x}_3 = 17.88 \) and \( s_3 = 3.74 \):\[CI = (19.84 - 17.88) \pm 1.341 \cdot \sqrt{\frac{3.07^2}{15} + \frac{3.74^2}{15}}\]\[CI = 1.96 \pm 1.341 \cdot \sqrt{\frac{9.4249}{15} + \frac{13.9876}{15}}\]\[CI = 1.96 \pm 1.341 \cdot 1.401\]\[CI = 1.96 \pm 1.878\]\[CI = (0.082, 3.838)\]

05

Calculate CI for 渭鈧� - 渭鈧�

Finally, calculate for \( \mu_2 - \mu_3 \):\[CI = (19.32 - 17.88) \pm 1.341 \cdot \sqrt{\frac{3.62^2}{15} + \frac{3.74^2}{15}}\]\[CI = 1.44 \pm 1.341 \cdot \sqrt{\frac{13.1044}{15} + \frac{13.9876}{15}}\]\[CI = 1.44 \pm 1.341 \cdot 1.401\]\[CI = 1.44 \pm 1.878\]\[CI = (-0.438, 3.318)\]

06

Interpretation of Results

The confidence intervals suggest that:- The difference in average self-esteem measure between competence and social acceptance is likely between \(-1.295\) and \(2.335\), indicating no significant difference.- The difference between competence and attractiveness is between \(0.082\) and \(3.838\), suggesting competence might have a higher impact.- The difference between social acceptance and attractiveness is between \(-0.438\) and \(3.318\), also not clearly indicating a significant difference.

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Key Concepts

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

Confidence Intervals

Confidence intervals are a vital concept in statistics, offering a range within which we can be 'confident' that the true population parameter lies. When looking at differences between two group means, a confidence interval tells us how much one mean might differ from another. For instance, in the self-esteem study, we calculate confidence intervals to determine the potential range of differences between means of different self-esteem indices.
The calculations use the mean and standard deviation of each group, along with a statistical measure known as the critical t-value. The result provides a range (or interval) within which the true difference of means may lie. It's important to note that an 85% confidence interval implies that, if we were to repeat the experiment many times, approximately 85% of the calculated intervals would contain the true difference in means.

• Offers a range of values for the mean difference
• Used to gauge the reliability of sample statistics
• Requires assumptions about the sample and test conditions to be accurate

Confidence intervals do not, however, provide a probability that the true value lies within the interval 鈥� they are not probabilities themselves but reflect the reliability based on the sample data.

Self-Esteem Study

Self-esteem is a complex psychological construct that is influenced by various factors. In this study, the researchers evaluated self-esteem among female undergraduates through different dimensions 鈥� competence, social acceptance, and physical attractiveness. Each of these dimensions is quantified as a random variable with respective point values.
The goal was not just to measure the level of self-esteem but to understand the variability and potential differences in influences that these three dimensions have on overall self-esteem. Such studies are critical as they can provide insights into what aspects of life significantly impact individuals' self-perception and how they relate to one another. This is particularly important in academic settings, where boosting self-esteem can lead to better academic and social outcomes.

• Examines multidimensional aspects of self-esteem
• Highly relevant for psychological and educational research
• Sheds light on which self-esteem factors might be more influential

Understanding these differences can guide interventions to enhance aspects of self-esteem that matter the most for personal development.

Random Variables

In statistics, a random variable is a quantity representing an outcome of a random phenomenon. In the self-esteem study, each dimension of self-esteem is treated as a random variable (x_1, x_2, x_3dots), each with its own distribution characterized by its mean and standard deviation.
Random variables provide a systematic way of describing uncertainty and variability in data. They enable researchers to use probability distributions to summarize how likely various outcomes are and to predict the potential range of these outcomes. For example, in the self-esteem study:- \(x_1\) represents competence,- \(x_2\) is social acceptance,- \(x_3\) denotes attractiveness.Each variable's distribution offers important context for understanding how much each factor contributes to the overall self-esteem in varying degrees among individuals in the sample.

• Describes variability in a systematic manner
• Helps in deriving meaningful insights from data
• Facilitates the calculation of confidence intervals and other statistical measures

Recognizing and understanding random variables enables a deeper comprehension of data tendencies and predictions in psychology and beyond.

Critical t-value

The critical t-value is a key component when calculating confidence intervals, especially with smaller sample sizes, which typically involve t-distribution. It's a threshold from the t-distribution table that reflects the desired confidence level 鈥� in this case, 85% 鈥� along with the sample's degrees of freedom (calculated as total sample size minus one).
In the self-esteem study, for a sample size of 15, the critical t-value used was approximately 1.341. This value is part of the calculation that determines the width of the confidence interval, reflecting how much the sample mean might deviate from the true population mean. Choosing the right t-value is crucial, as it affects the precision of the confidence interval and therefore the reliability of the conclusions drawn.

• Depends on confidence level and degrees of freedom
• Influences the width of confidence intervals
• Important for ensuring statistical accuracy in small sample studies

Correct application of the critical t-value paves the way for making confident, data-informed decisions in both scientific and practical contexts.