QuestionYou wish to test the following claim at a significance level of .
You obtain a sample of size in which there are 374 successful observations.
What is the test statistic for this sample? (Report answer accurate to two decimal places.)
test statistic
What is the -value for this sample? (Report answer accurate to four decimal places.)
p -value =
The -value is...
less than (or equal to)
greater than
This test statistic leáds to a decision to...
reject the null
accept the null
fail to reject the null
As such, the final conclusion is that...
There is sufficient evidence to warrant rejection of the claim that the population proportion is not equal to 0.54 .
There is not sufficient evidence to warrant rejection of the claim that the population proportion is not equal to 0.54 .
The sample data support the claim that the population proportion is not equal to 0.54 .
There is not sufficient sample evidence to support the claim that the population proportion is not equal to 0.54 .
Studdy Solution
STEP 1
What is this asking?
We're checking if a sample with **374** successes out of **707** trials gives us enough evidence to say the *true* success rate isn't **0.54**, with a significance level of **0.05**.
Watch out!
Don't mix up the sample proportion with the population proportion!
Also, remember that a two-tailed test means we're looking for differences in *either* direction from the assumed population proportion.
STEP 2
1. Calculate the Sample Proportion
2. Calculate the Standard Error
3. Calculate the Test Statistic
4. Calculate the p-value
5. Interpret the p-value and Make a Decision
STEP 3
Let's **dive in** by finding the sample proportion, often called *p-hat*.
It's simply the number of successes divided by the total number of trials.
In our case, we have **374** successes out of **707** trials.
STEP 4
So, .
This tells us that approximately **52.9%** of our sample were successes.
STEP 5
Next up, we need the **standard error**.
This tells us how much our sample proportion is likely to vary from the *true* population proportion.
The formula is , where is the assumed population proportion under the null hypothesis (**0.54** in our case) and is the sample size (**707**).
STEP 6
Plugging in our values, we get .
STEP 7
Now for the **star of the show**, the test statistic!
This tells us how far our sample proportion is from the assumed population proportion, measured in standard errors.
The formula is .
STEP 8
We've got all the pieces: , , and .
Let's plug them in! .
STEP 9
The **p-value** tells us the probability of observing a sample as extreme as ours (or even *more* extreme) if the null hypothesis were *actually* true.
Since this is a two-tailed test, we need to consider both tails of the distribution.
STEP 10
Using a z-table or calculator, we find that the area to the left of is approximately **0.2776**.
Since it's a two-tailed test, we double this value to get the p-value: .
STEP 11
Our p-value (**0.5552**) is *greater* than our significance level (**0.05**).
This means we *don't* have enough evidence to reject the null hypothesis.
STEP 12
Think of it like this: if the *true* population proportion *really was* **0.54**, it wouldn't be that unusual to get a sample like ours.
STEP 13
Test statistic: p-value: The p-value is greater than . This test statistic leads to a decision to fail to reject the null. As such, the final conclusion is that there is not sufficient sample evidence to support the claim that the population proportion is not equal to 0.54.
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