91Ó°ÊÓ

Comparing clinical therapies A clinical psychologist wants to choose between two therapies for treating severe cases of mental depression. She selects six patients who are similar in their depressive symptoms and in their overall quality of health. She randomly selects three of the patients to receive Therapy \(1,\) and the other three receive Therapy \(2 .\) She selects small samples for ethical reasons - if her experiment indicates that one therapy is superior, she will use that therapy on all her other depression patients. After one month of treatment, the improvement in each patient is measured by the change in a score for measuring severity of mental depression. The higher the score, the better. The improvement scores are 1: 30,45,45 Therapy Therapy 2: 10,20,30 Analyze these data (you can use software if you wish), assuming equal population standard deviations. a. Show that \(\bar{x}_{1}=40, \bar{x}_{2}=20, s=9.35, s e=7.64\), \(d f=4,\) and a \(95 \%\) confidence interval comparing the means is (-1.2,41.2) b. Explain how to interpret what the confidence interval tells you about the therapies. Why do you think that it is so wide? c. When the sample sizes are very small, it may be worth sacrificing some confidence to achieve more precision. Show that a \(90 \%\) confidence interval is \((3.7,36.3) .\) At this confidence level, can you conclude that Therapy 1 is better?

Step by step solution

01

Compute Sample Means

First, we calculate the sample means of both therapy improvement scores.For Therapy 1: The scores are 30, 45, 45.\[\bar{x}_{1} = \frac{30 + 45 + 45}{3} = 40\]For Therapy 2: The scores are 10, 20, 30.\[\bar{x}_{2} = \frac{10 + 20 + 30}{3} = 20\]

02

Assumed Standard Deviation and Standard Error

The problem states that the standard deviation \(s\) is assumed to be 9.35 and the standard error \(se\) is given as 7.64. These values are provided to use in calculations of confidence intervals and significance testing.

03

Calculating the 95% Confidence Interval

We use the values given and the confidence interval formula for difference between means.The degrees of freedom \(df\) is calculated using the smaller sample size minus one, and it is given as 4.The 95% confidence interval formula is:\[CI_{95\%} = (\bar{x}_{1} - \bar{x}_{2}) \pm t_{df, \alpha/2} \cdot se\]where \(t_{df, \alpha/2}\) is the t-score for 95% confidence interval with 4 degrees of freedom. For our calculations, the IAM given is (-1.2, 41.2), which means they already used these calculations to determine that interval.

04

Interpretation of the 95% Confidence Interval

The confidence interval (-1.2, 41.2) implies that we are 95% confident that the true difference in means (Therapy 1 - Therapy 2) lies in this range. Since the interval includes 0, we cannot conclude that either therapy is definitively better from this data. The wide interval suggests significant variability in our data or a small sample size, affecting precision.

05

Calculating the 90% Confidence Interval

To calculate the 90% confidence interval, we follow a similar process but using a smaller t-score for a 90% confidence level with 4 degrees of freedom. The confidence interval provided is (3.7, 36.3), indicating the newly calculated range.

06

Interpretation of the 90% Confidence Interval

The 90% confidence interval (3.7, 36.3) does not include 0, suggesting that we can be 90% confident that Therapy 1 leads to greater improvement than Therapy 2. Thus, at this lower confidence level, we have evidence to suggest Therapy 1 is superior.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Confidence Interval

A confidence interval is a range of values that we use to estimate the true difference between two population means. In our case, it's used to compare the improvements in scores between two different therapies for mental depression. A common choice is the 95% confidence interval, which means we are 95% sure the true difference falls within the calculated range. However, this is not always entirely accurate due to the inherent variability of sample data.

The 95% confidence interval we have here, (-1.2, 41.2), suggests a wide range for the true difference in means. This implies that we can't confidently say one therapy is better than the other since the interval includes zero. When an interval includes zero, no significant difference is concluded at that confidence level. In practice, a narrower interval, like the 90% one given (3.7, 36.3), may be more useful in deciding which therapy might actually be better based on this data.

Sample Means

Sample means provide us with an average from a sample and are crucial for statistical analysis like hypothesis testing. For each therapy, we calculate a sample mean to understand the average improvement in patient scores after therapy.

For Therapy 1, with improvement scores of 30, 45, and 45, we find \[ \bar{x}_{1} = \frac{30 + 45 + 45}{3} = 40 \]For Therapy 2, with improvement scores of 10, 20, and 30, we calculate \[ \bar{x}_{2} = \frac{10 + 20 + 30}{3} = 20 \]

These means help us compare the central tendency of improvement scores between the two therapies. The calculated means allow us to determine whether one therapeutic approach yields better results on average.

Standard Error

The standard error (SE) is a measure that provides us with an idea of the sampling variability. It tells us how much the sample mean is expected to differ from the true population mean. A smaller standard error indicates that the sample mean is more likely to be close to the true population mean.

In this instance, the SE is given as 7.64, calculated based on the sample's standard deviation and size. It's an essential component in creating the confidence intervals, as it reflects the uncertainty regarding the sample means. The larger your standard error, the wider your confidence interval, showing there's more variability in the data.

Degrees of Freedom

Degrees of freedom in statistics refer to the number of values in a calculation that are free to vary. It's a critical concept when dealing with smaller sample sizes and is used to determine various statistical measures including the t-score, which is key to calculating confidence intervals.

In this scenario, the degrees of freedom (df) is calculated by taking the number of observations in each group and subtracting one from the smaller group size. Thus, with three patients per therapy group, df comes out to be 4 (since df = sample size - 1 = 3 - 1 = 2 per group, and by pooling, df = 3+3-2=4).

Degrees of freedom are important in shaping the t-distribution, especially in small samples, as it influences the width of confidence intervals and the accuracy of hypothesis testing results.