Math Snap

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: $H_0: p_1 = p_2$, where $p_1$ is the proportion of males with at least one tattoo, and $p_2$ is the proportion of females with at least one tattoo.

STEP 4

Our alternative hypothesis is that the proportions are not the same.
Formally: $H_1: p_1 \ne p_2$.

STEP 5

We have 183 tattooed males out of 1100.
So, $\hat{p}_1 = \frac{183}{1100} \approx \textbf{0.166}$.
That's about 16.6% of the males surveyed.

STEP 6

We have 133 tattooed females out of 1000.
So, $\hat{p}_2 = \frac{133}{1000} = \textbf{0.133}$.
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

$$\hat{p} = \frac{\text{Total tattooed males + Total tattooed females}}{\text{Total males + Total females}} = \frac{183 + 133}{1100 + 1000} = \frac{316}{2100} \approx \textbf{0.150}$$ So, our pooled proportion is about 15.0%.

STEP 9

We'll use the z-test for comparing two proportions.
Here's the formula:
$$z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$$

STEP 10

Remember, $\hat{p}_1 \approx 0.166$, $\hat{p}_2 = 0.133$, $\hat{p} \approx 0.150$, $n_1 = 1100$, and $n_2 = 1000$.
Let's plug those in:
$$z = \frac{(0.166 - 0.133) - 0}{\sqrt{0.150(1 - 0.150)(\frac{1}{1100} + \frac{1}{1000})}} = \frac{0.033}{\sqrt{0.150 \cdot 0.850 \cdot (\frac{2100}{1100 \cdot 1000})}} \approx \frac{0.033}{0.015} \approx \textbf{2.20}$$ 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 $z = 2.20$ and $z = -2.20$.
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 ($\alpha = 0.05$).

STEP 14

This means we reject the null hypothesis.

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.