91Ó°ÊÓ

Reincarnation Suppose \(21 \%\) of all American teens (age 13-17 years) believe in reincarnation. (a) Bob and Alicia both obtain a random sample of 100 American teens and ask each participant to disclose whether they believe in reincarnation or not. Is "belief in reincarnation" qualitative or quantitative? Explain. (b) Explain why Bob's sample of 100 randomly selected American teens might result in 18 who believe in reincarnation, while Alicia's independent sample of 100 randomly selected American teens might result in 22 who believe in reincarnation. (c) Why is it important to randomly select American teens to estimate the population proportion who believe in reincarnation? (d) In a survey of 100 American teens, how many would you expect to believe in reincarnation? (e) Below is the histogram of the sample proportion of 1000 different surveys in which \(n=20\) American teens were asked to disclose whether they believed in reincarnation. Explain why the normal model should not be used to describe the distribution of the sample proportion. (f) What minimum sample size would you require in order for the distribution of the sample proportion to be modeled by the normal distribution?

Step by step solution

01

Understanding Type of Data

Determine whether 'belief in reincarnation' is qualitative or quantitative. Since it categorizes teens based on their belief, it is qualitative data.

02

Variability in Sample Results

Different random samples can yield different results for the number of teens who believe in reincarnation. This happens due to sampling variability. Even with the same sample size, each random sample can have different outcomes.

03

Importance of Random Selection

Random selection is crucial to avoid bias and ensure the sample accurately represents the population. It gives each teen an equal chance to be included, leading to more reliable estimates of the population proportion.

04

Expected Number Calculation

Using the given proportion, calculate the expected number of teens: The expected number is found by multiplying the proportion (0.21) by the sample size (100): \(0.21 \times 100 = 21\). Thus, 21 teens are expected to believe in reincarnation.

05

Inapplicability of the Normal Model for Small Sample

For a sample size of 20, the conditions for using the normal model (np≥10 and n(1-p)≥10) are not satisfied. With n=20 and p=0.21, \(np = 20 \times 0.21 = 4.2\) and \(n(1-p) = 20 \times 0.79 = 15.8\). Since 4.2 is less than 10, the normal model is inappropriate.

06

Minimum Sample Size

To ensure the normal model can be used, both np and n(1-p) must be at least 10. Solve for n using inequalities: \(0.21n \ge 10\) leading to \(n \ge 47.62\) \(0.79n \ge 10\) leading to \(n \ge 12.66\). Thus, the minimum sample size required is about 48, as it satisfies both conditions.

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Key Concepts

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

qualitative vs quantitative data

In statistics, data can be categorized as either qualitative or quantitative. Qualitative data refers to non-numeric information that categorizes or describes an attribute of a population. In the given exercise, 'belief in reincarnation' is qualitative data because it sorts teens into categories based on their belief.
Qualitative data is different from quantitative data, which involves numeric measurements. Examples of quantitative data could include the height, weight, or age of individuals. Understanding the distinction between these two types of data is crucial for choosing the right analysis methods.

sample size determination

Sample size determination is a critical step in statistical analysis. It involves deciding how many observations or elements are needed to represent a population accurately. In the exercise, Bob and Alicia each select a sample of 100 American teens to estimate the proportion who believe in reincarnation.
The size of the sample affects the reliability of the estimates. Larger samples tend to provide more stable and accurate estimates. The variability seen in the different results from Bob's and Alicia's samples (18 and 22 believers, respectively) highlights how different random samples can yield varying outcomes due to sampling variability.
Choosing the right sample size helps in ensuring that the results are representative of the larger population and that the conclusions drawn are valid.

normal distribution criteria

For the normal distribution to be an appropriate model for a sample proportion, specific criteria must be met. These criteria generally involve having a sufficiently large sample size such that both np and n(1-p) are greater than or equal to 10. This requirement ensures that the sampling distribution of the proportion can be approximated by a normal distribution.
In the exercise, for a sample size of 20, the conditions are not met. With n=20 and p=0.21, we get np=4.2 and n(1-p)=15.8, but np is less than 10. Therefore, the normal model should not be used to describe this distribution. The minimum sample size required for using the normal distribution in this scenario is approximately 48, as shown by the calculations ensuring both np and n(1-p) are at least 10.

population proportion estimation

Population proportion estimation involves determining the percentage of a population that possesses a particular characteristic. In the given exercise, the aim is to estimate the proportion of teens who believe in reincarnation. The known population proportion is 21%.
To estimate this proportion effectively, a random sample is selected, and the sample proportion is calculated. Random sampling ensures that each member of the population has an equal chance of being included, thereby minimizing bias.
The exercise expects 21 out of 100 teens to believe in reincarnation, calculated as 0.21 times 100. This helps to illustrate a principle in population proportion estimation: by multiplying the known proportion by the sample size, we can predict the number of individuals in the sample that manifest the trait of interest.