91Ó°ÊÓ

Testing a Random Number Generator. A random number generator is supposed to produce random numbers that are uniformly distributed on the interval from 0 to 1 . If this is true, the numbers generated come from a population with \(\mu=0.5\) and \(\sigma=0.2887\). A command to generate 100 random numbers gives outcomes with mean \(x=0.5635\). Assume that the population \(\sigma\) remains fixed. We want to test $$ \begin{aligned} &H_{0}: \mu=0.5 \\ &H_{a}: \mu \neq 0.5 \end{aligned} $$ a. Calculate the value of the \(z\) test statistic. b. Use Table C: Is \(z\) statistically significant at the \(5 \%\) level \((\alpha=0.05) ?\) c. Use Table C: Is \(z\) statistically significant at the \(1 \%\) level \((\alpha=0.01)\) ? d. Between which two Normal critical values \(z^{*}\) in the bottom row of Table \(C\) does \(z\) lie? Between what two numbers does the \(P\)-value lie? Does the test give good evidence against the null hypothesis?

Step by step solution

01

Calculate the Test Statistic z

To calculate the test statistic, we start by using the formula for the z-score: \[ z = \frac{x - \mu}{\frac{\sigma}{\sqrt{n}}} \]where \(x = 0.5635\), \(\mu = 0.5\), \(\sigma = 0.2887\), and \(n = 100\). Substitute the given values:\[ z = \frac{0.5635 - 0.5}{\frac{0.2887}{\sqrt{100}}} \]\[ z = \frac{0.0635}{0.02887} \]\[ z = 2.200 \] Therefore, the z-score is approximately 2.200.

02

Determine Significance at 5% Level

For a two-tailed test with \(\alpha = 0.05\), the critical z-values are approximately \(\pm 1.96\). The calculated z-value is 2.200, which is greater than 1.96. Therefore, it falls in the rejection region, making it statistically significant at the 5% significance level.

03

Determine Significance at 1% Level

For a two-tailed test with \(\alpha = 0.01\), the critical z-values are approximately \(\pm 2.58\). The calculated z-value is 2.200, which is less than 2.58. Therefore, it does not fall in the rejection region, making it not statistically significant at the 1% significance level.

04

Determine P-value Range and Evidence

The z-value of 2.200 lies between the critical values for \( \alpha = 0.05 \) (2.054) and \( \alpha = 0.01 \) (2.326), which means the P-value is between 0.02 and 0.05.Since the P-value is less than 0.05 but more than 0.01, the test provides evidence against the null hypothesis, but it is not strong enough to reject the null hypothesis at a 1% significance level.

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Key Concepts

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

z-score

A z-score is a statistic used in hypothesis testing to measure the number of standard deviations an element is from the mean. It is calculated with the formula:

• \[ z = \frac{x - \mu}{\frac{\sigma}{\sqrt{n}}} \]

where:

• \(x\) is the sample mean,
• \(\mu\) is the population mean,
• \(\sigma\) is the standard deviation of the population,
• \(n\) is the sample size.

The z-score helps determine how likely it is to get the observed sample mean if the null hypothesis is true. For example, a z-score of 2.200 means the sample mean is 2.200 standard deviations above the population mean. This indicates that the observed outcome is somewhat unusual compared to what's expected under the null hypothesis. Therefore, z-scores are critical in making conclusions based on hypothesis testing.

Significance Level

Significance level, often denoted as \(\alpha\), is a threshold used in hypothesis testing to determine whether an observed effect is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (Type I error).In typical scenarios, common significance levels are 5% (\(\alpha = 0.05\)) and 1% (\(\alpha = 0.01\)).

• If the calculated z-value falls within the critical region defined by the significance level, the results are considered statistically significant.
• A 5% significance level means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis.

Using these levels in practice, at \(\alpha = 0.05\), our calculated z-score of 2.200 suggests that our finding is significant (as 2.200 > 1.96), signifying substantial evidence against the null hypothesis.

P-value

The P-value is the probability of observing a test statistic at least as extreme as the one obtained, assuming the null hypothesis is true.

• A smaller P-value indicates stronger evidence against the null hypothesis.
• In the context of this exercise, the P-value is between 0.02 and 0.05.

This means there's a 2% to 5% probability that the observed mean could occur by random chance if the generator's output truly followed the uniform distribution. If the P-value is less than the significance level, the null hypothesis is rejected. Here, since the P-value is less than 0.05, our result is statistically significant at the 5% level, but not significant enough at 1%, since 0.02 < P-value < 0.05.

Random Number Generator

A Random Number Generator (RNG) is a tool designed to produce a sequence of numbers that lack any predictable pattern. The expectation is that these numbers follow a specific statistical distribution, such as uniform distribution, across a defined interval, often 0 to 1. The effectiveness of an RNG in hypothesis testing depends on its ability to output numbers that conform to the assumed distribution. If numbers deviate significantly from an expected distribution, the generator might not be functioning correctly. In evaluating an RNG using hypothesis testing, the z-score of data samples helps assess if observed variations are statistically significant. Here, the mean of generated numbers deviated from the expected 0.5, prompting the need for statistical testing to conclude the generator's reliability.

Uniform Distribution

Uniform distribution is a type of probability distribution where all outcomes are equally likely within the specified range. For a continuous uniform distribution between 0 and 1, the meaning is simple:

• All numbers between 0 and 1 have the same probability of occurrence.
• The mean \(\mu\) is 0.5,
• and the standard deviation \(\sigma\) is approximately 0.2887.

If a random number generator is presumed to be uniform, any deviation in its outputs, such as an observed mean different from 0.5, needs investigation through hypothesis testing.In this exercise, evaluating the uniform distribution's parameter estimate through generating 100 samples revealed incongruities (mean=0.5635), confirming the need for statistical tests, including z-score calculations, to validate or refute uniformity.