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The Monterey Bay Aquarium, founded in 1984 , is situated on the beautiful coast of Monterey Bay in the historic Cannery Row district. In 1985 , the aquarium began a survey program that involved randomly sampling visitors as they exit for the day. The survey includes visitor demographic information, use of social media, and opinions on their aquarium visit. In 2015, the survey included 356 visitors over age 65 , of which 52 used a mobile device such as an Android phone or iPad during their visit. \({ }^{11}\) (a) What is the margin of error of the large-sample \(95 \%\) confidence interval for the proportion of visitors over 65 who used a mobile device during their visit? (b) How large a sample is needed to get the common \(\pm 3\) percentage point margin of error? Use the 2015 sample as a pilot study to get \(p^{*}\).

Step by step solution

01

Calculate the sample proportion

To find the sample proportion of visitors over 65 who used a mobile device, divide the number of users by the total number of visitors over 65. So, \( p = \frac{52}{356} \approx 0.1461 \).

02

Determine the margin of error formula

The margin of error for a large-sample \( 95\% \) confidence interval is calculated using the formula: \( E = Z_{95\%} \times \sqrt{\frac{p(1-p)}{n}} \), where \( Z_{95\%} \approx 1.96 \) for \( 95\% \) confidence, \( p \) is the sample proportion, and \( n \) is the sample size.

03

Calculate the margin of error

Substitute the values into the formula: \( E = 1.96 \times \sqrt{\frac{0.1461 \times (1-0.1461)}{356}} \). Calculating this gives \( E \approx 0.0351 \), or about \( 3.51\% \).

04

Use the formula for desired sample size

For a margin of error (E) of ±3 percentage points (\(0.03\)), we'll use the formula: \( n = \frac{Z_{95\%}^2 \times p^*(1-p^*)}{E^2} \). Substitute \( p^* = 0.1461 \) and \( E = 0.03 \), yielding \( n = \frac{1.96^2 \times 0.1461 \times 0.8539}{0.03^2} \).

05

Calculate the required sample size

Perform the calculation in the previous step: \( n \approx \frac{1.96^2 \times 0.1461 \times 0.8539}{0.0009} \approx 505.86 \). Rounding up, a sample size of 506 is needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Margin of Error

The margin of error is a key concept in statistics that indicates the range within which the true proportion of a population is likely to fall. When conducting surveys like the one at Monterey Bay Aquarium, it's crucial to know how accurate the results are.

The margin of error depends on the confidence level you choose. In our example, the confidence level is 95%. This means we are 95% confident that the true proportion of visitors over 65 using a mobile device falls within the calculated range.

To find the margin of error, we use the formula:

• \[ E = Z_{95 ext{%}} \times \sqrt{\frac{p(1-p)}{n}} \]
• Here, \(E\) is the margin of error, \(Z_{95\%} \approx 1.96\) is the z-score for a 95% confidence level, \(p\) is the sample proportion, and \(n\) is the sample size.

In this case, after doing the math, the margin of error turned out to be about 3.51%. This means you can expect a variation of 3.51% above or below the calculated proportion in the true population.

Confidence Interval

A confidence interval provides a range of values that likely contains the population parameter. When the Monterey Bay Aquarium obtained survey results, they sought a 95% confidence interval for the proportion of visitors over 65 using mobile devices.

The confidence interval is constructed around a central value known as the sample proportion, which in this instance is approximately 0.1461. This represents the proportion of surveyed individuals who used a mobile device.

To construct the confidence interval, we add and subtract the margin of error from the sample proportion:

• Lower limit: \( p - E = 0.1461 - 0.0351 \approx 0.1110 \)
• Upper limit: \( p + E = 0.1461 + 0.0351 \approx 0.1812 \)

Thus, the 95% confidence interval ranges from approximately 11.10% to 18.12%. This interval embodies our 95% assurance that the true proportion of mobile device users among visitors over 65 years old falls within these limits.

Sample Size Calculation

Determining the right sample size is vital for obtaining accurate survey results. With the Monterey Bay Aquarium survey, the challenge was ensuring the margin of error remained within ±3 percentage points.

To find this sample size, we can use a specific formula:

• \[ n = \frac{Z_{95\%}^2 \times p^*(1-p^*)}{E^2} \]
• Here, \(n\) is the required sample size, \(p^* = 0.1461\) is the estimated proportion from the pilot study, \(Z_{95\%} = 1.96\), and \(E = 0.03\) is the desired margin of error.

After substituting the values, the calculated sample size is roughly 506 visitors.

By aiming for this sample size, the aquarium can be reasonably sure that their findings will accurately represent the broader visitor population, maintaining a margin of error of just ±3 percentage points.