91Ó°ÊÓ

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 \(x^{-} \bar{x}=0.4365\). 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 \(\mathrm{C}\) : is \(z\) statistically significant at the \(5 \%\) level \((\alpha=0.05)\) ? (c) Use Table \(\mathrm{C}\) : 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

Formula for z Test Statistic

The formula to calculate the z test statistic is \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]where \(\bar{x} = 0.4365\), \(\mu = 0.5\), \(\sigma = 0.2887\), and \(n = 100\).

02

Calculate Standard Error

Calculate the standard error using the formula \[ \frac{\sigma}{\sqrt{n}} = \frac{0.2887}{\sqrt{100}} = 0.02887 \]

03

Substitute Values into z Formula

Using the values calculated, substitute into the z formula: \[ z = \frac{0.4365 - 0.5}{0.02887} \]

04

Calculate z

Compute the value of the z statistic:\[ z = \frac{-0.0635}{0.02887} \approx -2.20 \]

05

Compare z with Critcal Values at α = 0.05

At \( \alpha = 0.05 \), the critical z values are approximately \(-1.96\) and \(1.96\). Since \(-2.20\) falls outside this range, it is statistically significant.

06

Compare z with Critical Values at α = 0.01

At \( \alpha = 0.01 \), the critical z values are approximately \(-2.58\) and \(2.58\). Since \(-2.20\) falls outside this range, it is not statistically significant.

07

Determine z Position and P-value Range

The calculated \(z\) value \(-2.20\) falls between the critical values \(-2.58\) and \(-1.96\). The P-value lies between \(0.01\) and \(0.05\). Since the P-value is between these numbers, there is moderate evidence against the null hypothesis.

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.

Random Number Generator

A random number generator (RNG) is a tool or program used to create numbers in such a way that each number has an equal likelihood of being chosen. These numbers are often used in simulations or lotteries, where fairness and unpredictability are crucial. In the context of the exercise, we deal with numbers that should ideally exhibit a uniform distribution between 0 and 1. This means each number within this interval is equally probable to be generated.

• A good RNG ensures diversity and lack of patterns, mirroring the nature of random events.
• The specific mean (\(\mu\)) expected here is 0.5, which corresponds to the midpoint of the uniform distribution.
• The standard deviation (\(\sigma\)) represents the variability in the results, calculated here as 0.2887.

While a perfect RNG is theoretically impossible for computers (which are deterministic), sophisticated algorithms attempt to mimic true randomness as closely as possible.

Normal Distribution

Normal distribution, commonly known as the bell curve, is pivotal in the world of statistics. It represents a perfectly symmetrical distribution of values centered around a mean (\(\mu\)). In a normal distribution:

• The peak of the curve is at the mean.
• The spread of the curve is determined by the standard deviation (\(\sigma\)).
• Approximately 68% of the data should lie within one standard deviation of the mean, about 95% within two, and around 99.7% within three.

In the context of z-tests, normal distribution is crucial as it allows for comparisons using standard scores. The calculated z-score tells you how many standard deviations away from the mean your sample lies. A normal distribution hence serves as a backdrop for determining the significance of the result through its bell-shaped characteristics.

P-value

The P-value, or probability value, is a fundamental concept in hypothesis testing. It quantifies how likely it is to observe the data you have (or more extreme) under the assumption the null hypothesis is true. The smaller the P-value, the greater the evidence against the null hypothesis.

• A P-value less than 0.05 is typically considered statistically significant, suggesting strong evidence against the null hypothesis.
• In the exercise, the P-value is found between 0.01 and 0.05, signaling moderate evidence against the null hypothesis.

Analyzing the P-value helps in deciding whether the observed data significantly deviates from the expected scenario if the null hypothesis were true.

Null Hypothesis

The null hypothesis (\(H_0\)) is a starting assumption in hypothesis testing, asserting no effect or difference exists. In our exercise:

• The null hypothesis posits that the mean of generated numbers is equal to the assumed mean of 0.5, indicating a true uniform distribution.
• Our tests aimed to refute or fail to refute this assumption based on the data's evidence.
• A null result would suggest that any observed differences are due to random sampling variability rather than significant deviation.

By contrasting the null hypothesis with an alternative hypothesis (\(H_a\)), which asserts a difference does exist, we use data to guide evidence-based decisions. In this exercise, a z-test examines the mean's discrepancy from the hypothesis, providing insights into the data's validity or randomness.