91Ó°ÊÓ

In spite of the potential safety hazards, some people would like to have an Internet connection in their car. A preliminary survey of adult Americans has estimated this proportion to be somewhere around . 30 (USA Today. May 1, 2009\()\). a. Use the given preliminary estimate to determine the sample size required to estimate the proportion of adult Americans who would like an Internet connection in their car to within .02 with \(95 \%\) confidence. b. The formula for determining sample size given in this section corresponds to a confidence level of \(95 \%\). How would you modify this formula if a \(99 \%\) confidence level was desired? c. Use the given preliminary estimate to determine the sample size required to estimate the proportion of adult Americans who would like an Internet connection in their car to within .02 with \(99 \%\) confidence.

Step by step solution

01

Determine Sample Size for 95% Confidence Level

Use the given preliminary estimate to determine the sample size. We have \(Z = 1.96\) for a \(95\%\) confidence level, \(p = 0.30\) and \(E = 0.02\). Plugging these values into the sample size formula, we get: \[n = (1.96)^2 * (0.30) * (1 - 0.30) / (0.02)^2\] which gives us \(n = 2168.2\). Since we cannot have a fractional sample size, we round up to get \(n = 2169\).

02

Modifying the Formula for 99% Confidence Level

To modify this formula for a \(99\%\) confidence level, replace \(Z\) with the corresponding value for a \(99\%\) confidence level. For a \(99\%\) confidence level, \(Z = 2.576\). So the formula becomes :\[n = (2.576)^2 * p * (1 - p) / E^2\]

03

Determine Sample Size for 99% Confidence Level

Using the modified formula, we plug in the given values \(p = 0.30\) and \(E = 0.02\). So,\[n = (2.576)^2 * (0.30) * (1 - 0.30) / (0.02)^2\] which gives us \(n = 3777.62\). Rounding up, we get \(n = 3778\).

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values that is used to estimate the true value of a population parameter.
In this case, we're talking about estimating a proportion, such as the percentage of people who want an Internet connection in their car. The confidence interval gives us a range, and by stating a confidence level (like 95% or 99%), we're expressing how confident we are that the true proportion falls within this range.
It's like saying, "Given our data, we are 95% certain that the true proportion is between X and Y." Here's a little more on how this works:

• A 95% confidence interval means that if we repeated our study 100 times, we'd expect the true parameter to be in our calculated interval about 95 times.
• The size of the interval depends on the variability in the data and the size of our sample: larger samples typically yield narrower intervals.
• Changing the confidence level affects the width of the interval: higher confidence requires a wider interval.

Confidence intervals are crucial in survey research because they help us understand how much sampling error there might be in our estimate.

Proportion Estimation

Proportion estimation involves determining the proportion of a population that exhibits a certain characteristic, such as wanting internet access in a car.
In our example, we use the preliminary estimate that 30% of people want this feature. This is the sample proportion (\( p \)) and it serves as a starting point for calculating the required sample size to achieve a desired level of precision.
Here's some important point to consider:

• The sample proportion helps us estimate the population proportion, which is the true proportion we want to know.
• Accurate estimation involves not just guessing the correct fraction but also understanding the uncertainty around that guess, often expressed in terms of confidence intervals.
• Using a sample mean or proportion in calculations requires us to understand its potential variability, often handled by the standard error of the proportion.

In summary, understanding how to estimate proportions helps researchers design their surveys correctly, ensuring that they have enough samples to make accurate inferences about the whole population.

Survey Sampling

Survey sampling is a technique used to collect data from a subset of a population.
The information gained from the sample is intended to represent the entire population adequately. Here's a relatable explanation that helps clarify the process:

• When conducting a survey, researchers collect data from just a few people instead of everyone in the population, saving time and resources.
• A well-chosen sample should represent the diversity and characteristics of the total population.
• The sample size, as seen in our exercise, is critical because it influences both the accuracy and precision of our survey outcomes.
• The variability among the surveyed individuals must be accounted for, requiring statistical tools like confidence intervals to provide complete insights.

Survey sampling essentially allows us to gather valuable information efficiently.
When done correctly, it guides decision-making and policy development, all while providing a snapshot of the larger group.

Statistical Significance

Statistical significance is a key concept in research used to determine whether the results obtained in a study are likely due to a real effect or simply due to random chance.
It's commonly used in hypothesis testing and helps decide whether to reject or accept a hypothesis.
In the context of our exercise on sample size determination, understanding statistical significance includes:

• The idea that a result is statistically significant if the likelihood of it being due to random variation is below a certain threshold (often 5%, which corresponds to a 95% confidence level).
• It helps researchers assess whether their sample provides strong enough evidence to generalize findings to the entire population.
• A larger sample size can increase the statistical power of a study, making it more likely to detect true effects if they exist.

Statistical significance ensures that the conclusions drawn from data aren't just "noise."
It aims to provide a scientific basis for understanding the likelihood of observing an effect, given the chance of random error.