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Chapter 6: Problem 92

Checking for survey errors One way of checking the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known facts about the population. About 12\(\%\) of American adults identify themselves as black. Suppose we take an SRS of 1500 American adults and let \(X\) be the number of blacks in the sample. (a) Show that \(X\) is approximately a binomial random variable. (b) Check the conditions for using a Normal approximation in this setting. (c) Use the Normal approximation to find the probability that the sample will contain between 165 and 195 blacks. Show your work.

Short Answer

Expert verified
0.8069 probability that the sample will contain between 165 and 195 blacks.

Step by step solution

01

Validate Binomial Random Variable

We are conducting a survey of 1500 adults to check how many identify as black. This can be seen as a sequence of Bernoulli trials.1. **Fixed Number of Trials (n):** The sample size is 1500, so we have 1500 trials.2. **Two Outcomes per Trial:** Each person either identifies as black or not.3. **Independent Trials:** The choice of one individual is independent of another.4. **Constant Probability (p):** The probability of identifying as black is 12\(\%\), or 0.12, for each trial. Given these conditions, **X** can be modeled as a binomial random variable **B(n = 1500, p = 0.12)**.
02

Normal Approximation Check

We use the Normal approximation for the binomial distribution when both **np** and **n(1-p)** are greater than or equal to 10.1. **Calculate np:** \(np = 1500 \times 0.12 = 180\)2. **Calculate n(1-p):** \(n(1-p) = 1500 \times 0.88 = 1320\)Both values exceed 10, so it is appropriate to use the Normal approximation.
03

Convert to Normal Variables

To apply the Normal approximation, we calculate the mean \(\mu\) and standard deviation \(\sigma\) of the binomial distribution.1. **Mean (\(\mu\)):** \(\mu = np = 180\)2. **Standard Deviation (\(\sigma\)):** \(\sigma = \sqrt{np(1-p)} = \sqrt{180 \times 0.88} \approx 11.49\)Thus, we approximate **X** with \(N(\mu = 180, \sigma \approx 11.49)\).
04

Calculate Desired Probability

We need to find the probability that the number of blacks in our sample, **X**, is between 165 and 195.Using the Normal approximation:1. Convert 165 and 195 to z-scores: - \(z_{165} = \frac{165 - 180}{11.49} \approx -1.30\) - \(z_{195} = \frac{195 - 180}{11.49} \approx 1.30\)2. Use z-tables or a calculator to find the probability between these z-scores: - \(P(-1.30 < Z < 1.30)\) - This corresponds to approximately 0.8069.So, the probability that the sample contains between 165 and 195 blacks is about 0.8069.

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Key Concepts

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

Binomial Distribution
The binomial distribution is a fundamental concept in statistics that models situations where there are two possible outcomes. In this context, the binomial distribution helps us understand survey sampling situations, like determining how many people out of a sample of 1500 adults identify as black. For a scenario to be modeled as a binomial distribution, four key criteria must be met:
  • There is a fixed number of trials. In our case, it is the 1500 adults surveyed.
  • Each trial has two possible outcomes, analogous to 'success' and 'failure.' Here, it would be identifying as black or not.
  • Trials are independent, meaning one person's response doesn't influence another's.
  • The probability of "success" remains constant across trials. Here, there's a 12% chance an individual will identify as black.
This context is ideal for a binomial distribution, represented as \(B(n = 1500, p = 0.12)\). Understanding these points makes it easier to model and predict outcomes in survey-based research.
Normal Approximation
Normal approximation is a practical approach to simplifying calculations in binomial distributions, especially for large sample sizes. For our binomial distribution with 1500 trials, calculating probabilities directly can be complex. That's where normal approximation becomes useful. To use normal approximation effectively:
  • Calculate \(np\), which must be 10 or more. For our survey, \(np = 1500 \times 0.12 = 180\).
  • Calculate \(n(1-p)\), which should also be 10 or more. Here, \(n(1-p) = 1500 \times 0.88 = 1320\).
Both values vastly exceed 10, confirming it's appropriate to approximate the binomial distribution as a normal distribution. This method enables us to simplify calculations by using normal distribution properties to estimate binomial probabilities.
Probability Calculation
Once we establish that our binomial distribution can be approximated by a normal distribution, we can proceed with probability calculations more straightforwardly.We first determine the mean and standard deviation for the normal distribution:
  • Mean ( \(\mu\) ) is calculated as \(np = 180\).
  • Standard deviation ( \(\sigma\) ) is determined by the formula \(\sigma = \sqrt{np(1-p)} = \sqrt{180 \times 0.88} \approx 11.49\).
Now, we need to find the probability that the number of blacks in the sample, \(X\), is between 165 and 195. Converting these to z-scores:
  • \(z_{165} = \frac{165 - 180}{11.49} \approx -1.30\)
  • \(z_{195} = \frac{195 - 180}{11.49} \approx 1.30\)
We then lookup these z-scores in a standard normal distribution table or use a calculator to find the probability between them: \(P(-1.30 < Z < 1.30)\), approximately 0.8069. This reveals an 80.69% chance that the number of individuals identifying as black in the sample falls within this range.
Survey Sampling Errors
Survey sampling errors are inherent challenges faced when conducting surveys, as no sample perfectly represents the population. These errors could stem from various factors such as undercoverage, where certain segments of the population are not adequately represented. Understanding survey sampling errors is crucial:
  • Nonresponse bias occurs when individuals chosen for the survey don't participate, potentially skewing results.
  • Undercoverage happens if specific groups are systematically omitted or underrepresented in the sample.
  • Random sampling errors are differences between the sample and the population simply due to the randomness of sampling.
To mitigate these errors, researchers can compare sample data with known population characteristics. Doing so helps identify discrepancies or biases, ensuring conclusions drawn from survey data are more accurate and reliable. Proper sampling techniques and awareness can significantly reduce these potential errors, leading to more trustworthy survey results.

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Most popular questions from this chapter

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