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A company executive claims that employees in his industry get 100 junk emails per day. To further investigate this claim, the tech department of the company conducts a study. The executive selects a random sample of 10 employees and records the number of junk emails they received that day. Here are the data: 125, 101, 109, 94, 122, 92, 119, 90, 118, 122.

The tech department would like to determine if the data provide convincing evidence that the true mean number of junk emails received this day by employees of this company differs from 100. What are the appropriate hypotheses?

A. [tex]$H_0: p=100$[/tex] versus [tex]$H_a: p \neq 100$[/tex], where [tex]$p$[/tex] is the true proportion of junk emails received this day by employees of this company.

B. [tex]$H_0: p=100$[/tex] versus [tex]$H_a: p\ \textgreater \ 100$[/tex], where [tex]$p$[/tex] is the true proportion of junk emails received this day by employees of this company.

C. [tex]$H_0: \mu=100$[/tex] versus [tex]$H_a: \mu \neq 100$[/tex], where [tex]$\mu$[/tex] is the true mean number of junk emails received this day by employees of this company.

D. [tex]$H_0: \mu=100$[/tex] versus [tex]$H_a: \mu\ \textgreater \ 100$[/tex], where [tex]$\mu$[/tex] is the true mean number of junk emails received this day by employees of this company.



Answer :

GinnyAnswer
Given the question, we need to determine the appropriate hypotheses for testing whether the true mean number of junk emails received by employees differs from 100. Let's carefully analyze the components of the problem:

1. Null Hypothesis ([tex]\( H_0 \)[/tex]):
- The null hypothesis is a statement that there is no effect or no difference, and it represents the claim being tested. Here, the executive claims that the true mean number of junk emails received per day by employees is 100. Therefore, the null hypothesis is that the mean number of junk emails ([tex]\( \mu \)[/tex]) is 100.
- Mathematically, this is represented as [tex]\( H_0: \mu = 100 \)[/tex].

2. Alternative Hypothesis ([tex]\( H_a \)[/tex]):
- The alternative hypothesis represents what we are trying to find evidence for. It is what we consider if the null hypothesis is rejected. We want to test if the true mean number of junk emails received per day is not equal to 100.
- Mathematically, this is represented as [tex]\( H_a: \mu \neq 100 \)[/tex].

Given these points, the correct pair of hypotheses should focus on [tex]\( \mu \)[/tex] (the true mean number of junk emails):

- [tex]\( H_0: \mu = 100 \)[/tex] versus [tex]\( H_a: \mu \neq 100 \)[/tex], where [tex]\( \mu \)[/tex] is the true mean number of junk emails received by employees of the company.

Hence, the appropriate hypotheses for this scenario are:

[tex]\[ H_0: \mu = 100 \quad \text{versus} \quad H_a: \mu \neq 100, \][/tex]
where [tex]\( \mu \)[/tex] is the true mean number of junk emails received this day by employees of this company.