91Ó°ÊÓ

The level of monoamine oxidase (MAO) activity in blood platelets \((\mathrm{nm} / \mathrm{mg}\) protein/ \(\mathrm{h})\) was determined for each individual in a sample of 43 chronic schizophrenics, resulting in \(\bar{x}=2.69\) and \(s_{1}=2.30\), as well as for 45 normal subjects, resulting in \(\bar{y}=6.35\) and \(s_{2}=4.03\). Does this data strongly suggest that true average MAO activity for normal subjects is more than twice the activity level for schizophrenics? Derive a test procedure and carry out the test using \(\alpha=.01\). [Hint: \(H_{0}\) and \(H_{\mathrm{a}}\) here have a different form from the three standard cases. Let \(\mu_{1}\) and \(\mu_{2}\) refer to true average MAO activity for schizophrenics and normal subjects, respectively, and consider the parameter \(\theta=2 \mu_{1}-\mu_{2}\). Write \(H_{0}\) and \(H_{\mathrm{a}}\) in terms of \(\theta\), estimate \(\theta\), and derive \(\hat{\sigma}_{\hat{\theta}}\) ("Reduced Monoamine Oxidase Activity in Blood Platelets from Schizophrenic Patients," Nature, July 28, 1972: 225-226).]

Step by step solution

01

Define Hypotheses

First, set up the hypotheses based on the given information. Let \( \mu_1 \) and \( \mu_2 \) represent the true average MAO activity for schizophrenics and normal subjects, respectively. Define \( \theta = 2\mu_1 - \mu_2 \). The null hypothesis \( H_0 \) is \( \theta = 0 \), which suggests there is no significant difference in MAO activity that is twice for normal subjects compared to schizophrenics. The alternative hypothesis \( H_a \) is \( \theta < 0 \), indicating that MAO activity in normal subjects is more than twice that in schizophrenics.

02

Estimate \( \theta \)

Use the sample means to estimate \( \theta \). Substitute the sample statistics as follows: \( \hat{\theta} = 2\bar{x} - \bar{y} \). Using the given values, \( \bar{x} = 2.69 \) for schizophrenics and \( \bar{y} = 6.35 \) for normal subjects, so \( \hat{\theta} = 2\times 2.69 - 6.35 = 5.38 - 6.35 = -0.97 \).

03

Calculate Standard Error of \( \hat{\theta} \)

The standard error for \( \hat{\theta} \) is given by \( \hat{\sigma}_{\hat{\theta}} = \sqrt{4\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \). Substitute the given statistics: \( s_1 = 2.30 \), \( n_1 = 43 \), \( s_2 = 4.03 \), \( n_2 = 45 \). Thus, \( \hat{\sigma}_{\hat{\theta}} = \sqrt{4\times\frac{2.30^2}{43} + \frac{4.03^2}{45}} = \sqrt{0.4934 + 0.3600} = \sqrt{0.8534} = 0.9238 \).

04

Z-Test for \( \theta \)

Perform a Z-test to determine if \( \theta < 0 \). The test statistic \( Z \) is given by \( Z = \frac{\hat{\theta} - 0}{\hat{\sigma}_{\hat{\theta}}} \). Substitute the values: \( Z = \frac{-0.97 - 0}{0.9238} = \frac{-0.97}{0.9238} = -1.0504 \).

05

Decision at \( \alpha = 0.01 \)

Compare the calculated Z statistic to the critical value from the standard normal distribution. For \( \alpha = 0.01 \) (one-tailed test), the critical value is approximately \(-2.33\). Since \(-1.0504 > -2.33\), we fail to reject the null hypothesis \( H_0 \). This suggests there is no significant evidence at the 0.01 level that the true average MAO activity for normal subjects is more than twice that of schizophrenics.

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Key Concepts

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

Null and Alternative Hypotheses

In statistical hypothesis testing, establishing clear null and alternative hypotheses is crucial as they set the foundation for the entire testing process. The null hypothesis, denoted as \( H_0 \), is a statement of no effect or no difference. It serves as the default or starting assumption. Conversely, the alternative hypothesis, \( H_a \), represents what we aim to support. It is a statement that indicates the presence of an effect or difference.

Using the given exercise, we had to define hypotheses in a slightly unconventional way. We defined \( \theta = 2\mu_1 - \mu_2 \), where \( \mu_1 \) and \( \mu_2 \) are the true average MAO activities for schizophrenics and normal subjects, respectively. Our null hypothesis \( H_0 \) was \( \theta = 0 \), indicating no difference in expected activity levels. The alternative hypothesis \( H_a \) was \( \theta < 0 \), suggesting that the MAO activity in normal subjects is more than twice that in schizophrenics.

Z-Test

A Z-test is a type of statistical test that determines if there is a significant difference between the means of two groups when the variance is known and the sample size is large (typically \( n > 30 \)). It is appropriate when the data is approximately normally distributed and when the population standard deviation is known or can be accurately estimated from the sample.

In our exercise, we performed a Z-test using the parameter \( \theta \) to determine if the average MAO activity for normal subjects is more than twice that of schizophrenics. We computed the test statistic \( Z \) by dividing the estimated difference, \( \hat{\theta} \), by its standard error \( \hat{\sigma}_{\hat{\theta}} \).
These calculations help assess whether observed data significantly deviate from the null hypothesis, indicating if \( \theta < 0 \) as per our defined alternative hypothesis.

Standard Error

The standard error is a measure that indicates the amount of variation or dispersion in a set of sample means. It estimates the standard deviation of the sample mean distribution, reflecting how much the estimated mean \( \hat{\theta} \) could vary if the sampling were repeated.

In the problem at hand, the computation of the standard error \( \hat{\sigma}_{\hat{\theta}} \) was crucial. We used:

• The formula \( \hat{\sigma}_{\hat{\theta}} = \sqrt{4\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \), which accounts for the variances of both schizophrenics and normal subjects.
• The sample sizes \( n_1 \) and \( n_2 \).
• The standard deviations \( s_1 \) and \( s_2 \).

A smaller standard error would imply higher precision of our mean estimates, affecting the reliability of the Z-test's outcome.

Significance Level

The significance level, often denoted as \( \alpha \), is a threshold set by researchers to decide whether to reject the null hypothesis. It quantifies the probability of rejecting the null hypothesis when it is actually true, commonly set at 0.05 or 0.01.

In our hypothesis testing scenario, the significance level was defined at \( \alpha = 0.01 \). This means there's a 1% risk of concluding that the true average MAO activity for normal subjects is more than twice that for schizophrenics when it is not. The critical value corresponding to this significance level is approximately -2.33 for a one-tailed test, which we compared with our calculated Z-statistic.

Using this level allowed us to maintain rigorous standards for decision-making in our test, ensuring reliable outcomes while considering the possibility of a Type I error.