Supposed Wall Street securities firms claim that they paid out year end bonuses of an average of 95,000 per employee last year. we take a sample of employee at the ASBE securities firm to see whether the average year-end bonus is less than the report mean of 95,000 for the populations. ### Hypothesis Testing of Year-End Bonuses**Scenario:**Suppose Wall Street securities firms claim that they paid out year-end bonuses of an average of $95,000 per employee last year. We take a sample of employees at the ASBE securities firm to see whether the average year-end bonus is less than the reported mean of $95,000 for the population.**Objective:**You wish to test the following claims at a significance level of $\alpha = 0.05$:- $H_0$: $\mu = \$95,000$- $H_a$: $\mu < \$95,000$**Assumptions:**- The population is normally distributed.- The population standard deviation is known to be 1800.- A sample mean of $94,575 is obtained from a sample size of 42.**Questions:**a) **What is the test statistic for this sample (use at least 3 decimal places)?**To calculate the test statistic (z):\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]Where:- $\bar{x} = 94575$- $\mu = 95000$- $\sigma = 1800$- $n = 42$Provide the detailed step-by-step computation.b) **What is the p-value for this sample?** (use Statcrunch)Use a statistical software like Statcrunch to find the exact p-value for the calculated test statistic.c) **Evaluate the strength of the evidence and state the conclusion (in the context of this question).**- Compare the p-value to the significance level $\alpha = 0.05$.- Interpret whether to reject or fail to reject the null hypothesis.- State the practical conclusion regarding the average year-end bonuses at ASBE securities firm. Provide the understanding in plain language for better comprehension.### Explanation of Graphs/Diagrams:There are no graphs or diagrams provided in this scenario. ---This example illustrates the process of hypothesis testing, emphasizing the steps from establishing hypotheses, calculating test statistics, determining p-values, and making informed conclusions based on statistical evidence.

(a). Test statistic z: The investigator is specially interested to test whether the average year-end…

Answered: Supposed Wall Street securities firms…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Supposed Wall Street securities firms claim that they paid out year end bonuses of an average of 95,000 per employee last year. we take a sample of employee at the ASBE securities firm to see whether the average year-end bonus is less than the report mean of 95,000 for the populations.

### Hypothesis Testing of Year-End Bonuses

**Scenario:**
Suppose Wall Street securities firms claim that they paid out year-end bonuses of an average of $95,000 per employee last year. We take a sample of employees at the ASBE securities firm to see whether the average year-end bonus is less than the reported mean of $95,000 for the population.

**Objective:**
You wish to test the following claims at a significance level of $\alpha = 0.05$:

- $H_0$: $\mu = \$95,000$
- $H_a$: $\mu < \$95,000$

**Assumptions:**
- The population is normally distributed.
- The population standard deviation is known to be 1800.
- A sample mean of $94,575 is obtained from a sample size of 42.

**Questions:**

a) **What is the test statistic for this sample (use at least 3 decimal places)?**

To calculate the test statistic (z):
\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]

Where:
- $\bar{x} = 94575$
- $\mu = 95000$
- $\sigma = 1800$
- $n = 42$

Provide the detailed step-by-step computation.

b) **What is the p-value for this sample?** (use Statcrunch)

Use a statistical software like Statcrunch to find the exact p-value for the calculated test statistic.

c) **Evaluate the strength of the evidence and state the conclusion (in the context of this question).**

- Compare the p-value to the significance level $\alpha = 0.05$.
- Interpret whether to reject or fail to reject the null hypothesis.
- State the practical conclusion regarding the average year-end bonuses at ASBE securities firm. Provide the understanding in plain language for better comprehension.

### Explanation of Graphs/Diagrams:
There are no graphs or diagrams provided in this scenario.

---

This example illustrates the process of hypothesis testing, emphasizing the steps from establishing hypotheses, calculating test statistics, determining p-values, and making informed conclusions based on statistical evidence.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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