Math Snap

STEP 1

What is this asking?
Out of 240 young cell phone owners, 108 have Androids.
We need to estimate how common Android phones are among this age group and see if it challenges an ad's claim of 38% Android ownership.
Watch out!
Don't mix up the number of people surveyed with the number who own Androids.
Also, remember that a confidence interval gives a range of plausible values, not just a single number.

STEP 2

1. Calculate the Point Estimate
2. Construct the Confidence Interval
3. Evaluate the Advertisement's Claim

STEP 3

Let's dive in!
We're given that $108$ out of $240$ young cell phone owners have Androids.
To get a point estimate for the proportion, we simply divide the number of Android owners by the total number surveyed.

STEP 4

So, our calculation is $\frac{108}{240}$.
This simplifies to $0.45$.
This means our point estimate is $0.45$, or \textbf{45%}.

STEP 5

Now, let's build a \textbf{90% confidence interval}.
This interval will give us a range of plausible values for the true proportion of Android owners.
The formula for a confidence interval is: $\text{Point Estimate} \pm \text{Margin of Error}$.

STEP 6

The margin of error is calculated as: $\text{Critical Value} \cdot \text{Standard Error}$.
For a 90% confidence level, the critical value (from a Z-table) is approximately $1.645$.

STEP 7

The standard error is calculated as $\sqrt{\frac{p(1-p)}{n}}$, where $p$ is our point estimate ($0.45$) and $n$ is our sample size ($240$).
Plugging in the values, we get $\sqrt{\frac{0.45 \cdot (1-0.45)}{240}} \approx \sqrt{\frac{0.2475}{240}} \approx \sqrt{0.00103} \approx 0.032$.

STEP 8

Now, we can calculate the margin of error: $1.645 \cdot 0.032 \approx 0.053$.

STEP 9

Finally, let's construct the confidence interval: $0.45 \pm 0.053$.
This gives us a range of $(0.45 - 0.053)$ to $(0.45 + 0.053)$, or $0.397$ to $0.503$.

STEP 10

The advertisement claims that $38\%$ of young cell phone owners have Androids.
Our \textbf{90% confidence interval} is from $0.397$ to $0.503$.

STEP 11

Since $0.38$ (or $38\%$) is not within our confidence interval, the interval does contradict the advertisement's claim.

The point estimate for Android phone ownership is $0.45$.
The 90% confidence interval is $(0.397, 0.503)$.
This interval contradicts the advertisement's claim of 38% Android ownership.