A recent survey of 8,000 high school students found that the mean price of a prom dress was $[tex]$195.00[/tex] with a standard deviation of $[tex]$12.00[/tex]. Alyssa thinks that her school is more fashion conscious and that students spent more than $[tex]$195.00[/tex]. She collected data from 20 people in her high school and found that the average price spent on a prom dress was $[tex]$208.00[/tex]. Which of the following are the correct null hypothesis and alternate hypothesis?
A. [tex]$H _0: \mu=195 ; H _{ a }: \mu\ \textgreater \ 195$[/tex]
B. [tex]$H_0: \mu \neq 195 ; H_a: \mu=208$[/tex]
C. [tex]$H _0: \mu=195 ; H _{ a }: \mu \neq 195$[/tex]
D. [tex]$H_0: \mu\ \textless \ 195 ; H_a: \mu \geq 208$[/tex]
Answer (2)
The null hypothesis states that the population mean is equal to $195: H 0 : μ = 195 .
The alternative hypothesis states that the population mean is greater than $195: 195"> H a : μ > 195 .
Therefore, the correct hypotheses are: 195"> H 0 : μ = 195 ; H a : μ > 195 .
The correct answer is: 195}"> H 0 : μ = 195 ; H a : μ > 195
Explanation
Understand the problem We need to identify the correct null and alternative hypotheses for Alyssa's hypothesis test. The null hypothesis represents the status quo, which in this case is that the mean price of prom dresses is $195. The alternative hypothesis represents Alyssa's belief that the mean price is greater than $195.
Define the null and alternative hypotheses The null hypothesis ( H 0 ) states that the population mean ( μ ) is equal to $195. This is written as: H 0 : μ = 195 The alternative hypothesis ( H a ) states that the population mean ( μ ) is greater than $195. This is written as: 195"> H a : μ > 195
State the final answer Therefore, the correct null and alternative hypotheses are: 195"> H 0 : μ = 195 ; H a : μ > 195
Examples
In hypothesis testing, we often want to determine if there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. For example, a company might want to test if a new marketing campaign has increased sales. The null hypothesis would be that the campaign has no effect, while the alternative hypothesis would be that the campaign has increased sales. By collecting data and performing a statistical test, the company can determine if there is enough evidence to reject the null hypothesis and conclude that the campaign was successful.
The correct hypotheses are the null hypothesis, which states that the mean price of a prom dress is 195 : H_0: \mu = 195$, and the alternative hypothesis that the mean price is greater than 195 : H_a: \mu > 195 . T h ere f ore , t h e an s w erc h o i ce i s A . H_0: \mu=195 ; H_a: \mu > 195$.
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