IHP 525 Quiz Two: Telephone Survey Uses A Random Digit Dial

Ihp 525 Quiz Twoa Telephone Survey Uses A Random Digit Dialing Machine

Ihp 525 Quiz Twoa Telephone Survey Uses A Random Digit Dialing Machine

A telephone survey employs a random digit dialing machine to contact subjects. The machine is expected to reach a live person 15% of the time. The assignment involves several probability and statistical questions, including modeling the probability of successful calls, estimating characteristics in a sample, and conducting hypothesis tests related to population parameters.

Paper For Above instruction

Introduction

Survey research is central to social sciences, marketing, and public health. Random digit dialing (RDD) is a common method to select representative samples in telephone surveys, where each call represents an independent event. This essay addresses multiple statistical concepts with applications rooted in survey procedures, probability theory, and inferential statistics, exemplified by diverse scenarios including contact success rates, trait prevalence, and height distributions. Understanding these concepts aids in interpreting survey data accurately and making valid statistical inferences about populations.

Probability of Successful Calls and Independent Events

The scenario involves a telephone survey where the RDD machine has a 15% success rate of reaching a live person. Each call can thus be modeled as a Bernoulli trial with p=0.15. It is standard in probability theory to assume that each call is an independent event, meaning the outcome of one call does not influence another. This assumption holds assuming the call process involves random selection from the population and no systematic bias in calling patterns.

Given this framework, the probability of achieving exactly two successful calls in two attempts follows a binomial probability distribution:

P(X=2)= (2 choose 2) (0.15)^2 (0.85)^0 = 1 0.0225 1 = 0.0225.

Similarly, the probability of having one success and one failure in any order involves the sum of the probabilities of two sequences:

P(1 success, 1 failure) = 2 (0.15)^1 (0.85)^1 = 2 0.15 0.85 = 2 * 0.1275 = 0.255.

These calculations illustrate the variability and likelihood of different outcomes in repeated independent call attempts, with the assumption of independence being valid under standard RDD procedures.

Estimating Trait Prevalence and Standard Deviation

In a simple random sample of size n=50, where the prevalence of a particular trait is 76.8%, the expected number of individuals exhibiting the trait can be estimated as:

Expected number = n p = 50 0.768 = 38.4.

The standard deviation of this estimate, considering the binomial distribution, is:

σ = √[n p (1-p)] = √[50 0.768 0.232] ≈ √[8.93] ≈ 2.99.

This standard deviation indicates the variability around the expected number and allows for constructing confidence intervals or assessing the precision of estimates based on the sample.

Probability of Contaminated Eggs in an Omelet

Linda learns that 1 in 6 eggs (approximately 16.67%) in the U.S. are contaminated with Salmonella. Assuming independent contamination within each egg and across eggs, the probability that an egg is contaminated (p) is 1/6.

If Linda uses three eggs to make an omelet, the probability that all three eggs are free of Salmonella is:

(1 - p)^3 = (5/6)^3 ≈ 0.5787.

Therefore, the probability that at least one egg is contaminated (the complement) is:

1 − 0.5787 ≈ 0.4213 or 42.13%.

This calculation highlights the risk associated with consuming multiple eggs and emphasizes the importance of proper food handling and testing.

Height Distribution among 10-year-old Boys

Heights of 10-year-old boys are modeled as a Normal distribution with mean μ=138 cm and standard deviation σ=7 cm. To find the proportion of boys shorter than 140 cm, we standardize the value:

z = (140 - 138) / 7 ≈ 0.2857.

Using standard Normal tables or computational tools, the cumulative probability up to z=0.2857 is approximately 0.611. Thus, about 61.1% of 10-year-old boys are shorter than 140 cm, demonstrating how normal distribution models can estimate proportions in populations based on sample statistics.

Estimating the Number of Health Problems and Normal Approximation

From a sample of 500 people, where the mean number of health problems per person is 2.3, it is reasonable to assume that the number of health problems per individual follows a normal distribution if the underlying distribution is approximately symmetric and the sample size is large. This assumption is consistent with the Central Limit Theorem, which states that the sampling distribution of the mean becomes normal as the sample size increases, typically n > 30.

Similarly, the sampling distribution of the sample mean (2.3) can also be assumed normal, enabling the use of parametric inference techniques.

Height Difference and Hypothesis Testing

A simple random sample of 18 male students has an average height of 70 inches, while the overall population mean male height is 69 inches with σ=2.8 inches. To assess whether the sample mean significantly differs, a two-sided hypothesis test can be performed:

  • Null hypothesis (H₀): μ = 69 inches
  • Alternative hypothesis (H₁): μ ≠ 69 inches

The test statistic (z) is calculated as:

z = (x̄ - μ₀) / (σ / √n) = (70 - 69) / (2.8 / √18) ≈ 1 / (2.8 / 4.24) ≈ 1 / 0.660 ≈ 1.515.

The p-value for a two-sided test is the probability of observing a z-value as extreme or more extreme: approximately 0.13, which exceeds the common significance level of 0.05. Therefore, there is insufficient evidence to conclude that the sample mean height differs significantly from the population mean.

The p-value indeed indicates the probability of observing the data or more extreme results assuming H₀ is true, reflecting the measure of evidence against the null hypothesis.

Conclusion

This exploration of survey data and statistical inference underscores the importance of understanding probability models, assumptions, and hypothesis testing. By modeling calls as independent Bernoulli trials, estimating the prevalence of traits, calculating food contamination risks, and applying normal distribution assumptions, researchers can make informed decisions and accurately interpret complex data. Proper statistical reasoning warrants caution, especially regarding assumptions about normality and independence, which underpin many common inferential procedures.

References

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