Question
Studdy Solution
STEP 1
What is this asking?
Is the percentage of tattooed males different from the percentage of tattooed females?
Watch out!
Don't mix up the groups (males vs. females) or their tattoo counts.
Also, remember we're looking for a *difference*, not whether one group has *more* tattoos than the other.
STEP 2
1. Set up the hypotheses
2. Calculate the sample proportions
3. Calculate the pooled proportion
4. Calculate the test statistic
5. Find the p-value
6. Make a decision
STEP 3
Our null hypothesis is that the proportions are the same.
Let's write this out formally: , where is the proportion of males with at least one tattoo, and is the proportion of females with at least one tattoo.
STEP 4
Our alternative hypothesis is that the proportions are *not* the same.
Formally: .
STEP 5
We have 183 tattooed males out of 1100.
So, .
That's about **16.6%** of the males surveyed.
STEP 6
We have 133 tattooed females out of 1000.
So, .
That's **13.3%** of the females surveyed.
STEP 7
We're assuming, for now, that the null hypothesis is true (the proportions are equal).
The pooled proportion is our best estimate of that *shared* proportion if they really were the same.
STEP 8
So, our pooled proportion is about **15.0%**.
STEP 9
We'll use the z-test for comparing two proportions.
Here's the formula:
STEP 10
Remember, , , , , and .
Let's plug those in:
Our test statistic is approximately **2.20**.
STEP 11
The p-value tells us the probability of observing a difference as extreme as the one we found (or even more extreme) if the null hypothesis were actually true.
STEP 12
Since this is a two-tailed test, we're looking for the area in *both* tails of the standard normal distribution beyond and .
Using a z-table or calculator, we find a p-value of approximately **0.028**.
STEP 13
Our p-value (**0.028**) is less than our significance level ().
STEP 14
This means we **reject the null hypothesis**.
STEP 15
There is statistically significant evidence to suggest that the proportion of males with at least one tattoo is different from the proportion of females with at least one tattoo.
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