Statistics Question

Overview

When we run a hypothesis test, we are testing whether or not a claim made is valid from a statistical standpoint. We break up our region into two regions, one is called the null hypothesis (H0), the other is called the alternative hypothesis (H1).  

There are some rules we have to follow when we make our hypotheses. First, the null hypothesis must always include some level of equality and to make this easier, the modern approach to writing hypotheses has us use = for all hypotheses. The alternative will then point to the appropriate region we desire to test and must be a strict inequality. This means the symbols will be >, <, or ?.   

Based on this, we can break our hypotheses up into three different sets:  

Left Tailed Hypotheses: H0:?=?0 vs H1:?

Right Tailed Hypotheses: vs H1:?>?0

Two Tailed Hypotheses: vs H1:???0

We determine our results from a hypothesis test by using the p-value. The p-value of our test is “the probability of seeing the result we saw, or more extreme, by random chance given that the null hypothesis is true.” If our p-value is small, we would say seeing something like this by random chance is very unlikely. If our p-value is larger, then it is more plausible to see something like this by random chance.  

The value of alpha, ?, is the probability we see a type I error, which means we reject the null hypothesis when the null hypothesis is in fact the truth. As the ones doing the research, we are able to select this value ourselves. We want to choose an alpha value that appropriately identifies the ramifications of making a mistake. If the results of our test have major outcomes associated with them, we may want to have a stricter (or smaller) value chosen for alpha. If the results of our test do not have major ramifications about them, we may want to choose a less strict (or larger) value for alpha. Common alpha values are ? = 0.1, 0.05, and 0.01.   

Our conclusion will be based on how our computed p-value relates to our value for alpha. If our p-value is less than alpha, we reject the null hypothesis. If our p-value is larger than alpha, we fail to reject the null hypothesis. In the infamous words of a brilliant man, “If p is small, reject them all.” 

Here is the process we follow when performing a hypothesis test:  

Step 1: Write out your hypotheses based on the words given in the problem  

Step 2: Determine your value of alpha  

Step 3: Compute your Z statistic  

Step 4: Compute your P-value  

Step 5: Make your conclusion

Now, let’s discuss how we compute our Z-value. If we know the population standard deviation, then our Z value will be z=x¯???n.

Now, let’s look at an example. Suppose the local car dealership claims their brand new mid-sized sedan gets at least 32 mpg on average with a standard deviation of 4 mpg. We want to test to see whether or not this claim is true. So, we collected a sample of 35 of these mid-sized sedans and found the average mpg was 30 mpg. Do we have enough evidence to dispute this claim?  

Step 1: H0:?=32mpg vs H1:? <32mpg

Step 2: ? = 0.05 (Chosen to be a touch stricter due to us wanting a little more certainty in our result before we make such big purchase) 

Step 3: =30?32435=?2.958

Step 4: P-value = 0.0015  

Step 5: Our p-value is smaller than our alpha value, so we will reject the null hypothesis and have statistical evidence to dispute the claim  

If we want to run a hypothesis test for the population proportion, we will follow a very similar construct as we did with the mean, except our Z-statistic calculation will change. Our hypotheses will be similar, except we will replace µ with p. Here are the hypotheses: 

Left Tailed Hypotheses: H0_p=p0 vs H1:p<p0

Right Tailed Hypotheses: vs H1:p>p0

Two Tailed Hypotheses: vs H1:p?p0

The Z value for the population proportion hypothesis test will be computed as follows: Z=p^?ppqn.

Let’s look at an example: Recent health statistics show that an estimated 12.5% of U.S. adults currently smoke cigarettes. We decide to test this claim, and survey 1500 adults and ask them whether they smoke or not. We have found that 195 of the 1500 (13%) claim to be smokers. Is there enough evidence to reject the claim made that the true proportion is 12.5%? 

Step 1: :p=0,125 vs :p? 0.125 

Step 2: ? = 0.1 (Chosen to be a little less strict as the outcome of our test may not have any direct impact on anyone or anything, but could be used to inform a decision maker if needed) 

Step 3: Z=p^?ppqn=0.13?0.1250.125?0.8751500=0.586

Step 4: P-value = 0.558 

Step 5: Our p-value is larger than our alpha value, so we will fail to reject the null hypothesis and do not have statistical evidence to dispute the claim

Instructions

For this discussion post, we are going to run a hypothesis test using the Z-distribution. Read the following:  

The average salary for a registered nurse in Las Vegas is claimed to be $82,000 with a standard deviation of $14,500. To determine if this information is accurate, we sampled 80 registered nurses working in Las Vegas and find their average salary to be $85,025. Test this at the ? = 0.05 level. 

Discussion Prompts

Answer the following questions in your initial post: 

What are the hypotheses based on the words given in the problem? 

What is our Z?

What is the P-value?

Based on your p-value and alpha, what conclusion will we make? Do we have evidence that the claim is false?

Are there any other variables that could potentially impact the salary of a registered nurse? What are some methods of sampling we can use to ensure we have a representative population.