Math / Data & Statistics

Questionwww-awuialeks.com Secret - Coo... 11/22/24 ATL... Home - Nort... Content ChatGPT Chicago B... Failed to ope... Online Bettin... (6) KaiCe. Homework * 4: $7(1,2,3,4) 8(2,3,4)$ Question 30 of 40 (1 point) I Question Attempt: 1 of 3 Jonath
Smart phone: Among 240 cell phone owners aged 18-24 surveyed, 108 said their phone was an Android phone. Perform the following.
Part 1 of 3 (a) Find a point estimate for the proportion of cell phone owners aged 18-24 who have an Android phone. Round the answer to at least three decimal places.
The point estimate for the proportion of cell phone owners aged $18-24$ who have android phone is 0.450 .
Part 2 of 3 (b) Construct a $90 \%$ confidence interval for the proportion of cell phone owners aged 18-24 who have an Android phone. Round the answer to at least three decimal places.
A $90 \%$ confidence interval for the proportion of cell phone owners aged 18-24 who have an Android phone is $0.397<p<0.503$
Part: $2 / 3$
Part 3 of 3 (c) Assume that an advertisement claimed that $38 \%$ of cell phone owners aged 18-24 have an Android phone. Does the confidence interval contradict this claim? claim?
The confidence interval $\square$ (Choose one) contradict the claim, because 0.38 (Choose one) contained in the confidence interval. Skip Part Check Save For Later Submit Assignm ( 2024 MaGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy Center I Acces:

Studdy Solution

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.

STEP 12

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.

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Studdy Solution

STEP 1

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.

STEP 12

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.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the Point Estimate2. Construct the Confidence Interval3. Evaluate the Advertisement's Claim

STEP 3

Let's **dive in**!We're given that 108108108 out of 240240240 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 108240\frac{108}{240}240108​.This simplifies to 0.450.450.45.This means our **point estimate** is 0.450.450.45, or \textbf{45%}.

STEP 5

STEP 6

The **margin of error** is calculated as: Critical Value⋅Standard Error\text{Critical Value} \cdot \text{Standard Error}Critical Value⋅Standard Error.For a 90% confidence level, the **critical value** (from a Z-table) is approximately 1.6451.6451.645.

STEP 7

STEP 8

Now, we can calculate the **margin of error**: 1.645⋅0.032≈0.0531.645 \cdot 0.032 \approx 0.0531.645⋅0.032≈0.053.

STEP 9

Finally, let's construct the **confidence interval**: 0.45±0.0530.45 \pm 0.0530.45±0.053.This gives us a range of (0.45−0.053)(0.45 - 0.053)(0.45−0.053) to (0.45+0.053)(0.45 + 0.053)(0.45+0.053), or 0.3970.3970.397 to 0.5030.5030.503.

STEP 10

The advertisement claims that 38%38\%38% of young cell phone owners have Androids.Our \textbf{90% confidence interval} is from 0.3970.3970.397 to 0.5030.5030.503.

STEP 11

Since 0.380.380.38 (or 38%38\%38%) is *not* within our confidence interval, the interval *does* contradict the advertisement's claim.

STEP 12

The point estimate for Android phone ownership is 0.450.450.45.The 90% confidence interval is (0.397,0.503)(0.397, 0.503)(0.397,0.503).This interval contradicts the advertisement's claim of 38% Android ownership.

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

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.

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

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 $1.645$ .

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

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 $0.503$ .

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

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.