Statistics & Probability Exam

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Questions and Answers

What does a random variable represent in relation to an experiment?

An average of the outcomes

A mathematical constant

A measure of the central tendency

A function that assigns a real number to each element in the sample space (correct)

What must the probability of each value of a random variable be?

Greater than 1

Exactly 0

Between 0 and 1, inclusive (correct)

Equal to the number of events

What is the sum of the probabilities of all values of a random variable supposed to equal?

The number of possible outcomes

1 (correct)

The mean value

Which of the following represents the mean of a discrete random variable?

The sum of the numbers divided by the total number of scores Signup and view all the answers

What values can the random variable X take when two coins are tossed?

0, 1, and 2 Signup and view all the answers

How is the mean of a discrete random variable calculated?

By summing the products of the values and their corresponding probabilities Signup and view all the answers

What is the meaning of the term 'probability distribution'?

It describes the likelihood of each outcome in a sample space Signup and view all the answers

If two coins are tossed, what does the random variable X represent?

The number of heads that will occur Signup and view all the answers

What happens to the raw score X when the z-score is positive?

It is above the mean. Signup and view all the answers

If a population has a mean of μ, what is the mean of the sampling distribution of the sample means?

Equal to μ Signup and view all the answers

What is the formula for the variance of the sampling distribution of the mean for an infinite population?

σ^2X̅ = σ^2 / n Signup and view all the answers

What is the mean of the sampling distribution of the sample means equal to?

The mean of the population Signup and view all the answers

What is the formula for the variance of the sampling distribution of the sample means for a finite population?

$σ^2/n (N - n / N - 1)$ Signup and view all the answers

What does the standard error of the mean represent in a sampling distribution?

The accuracy of the sample mean as an estimate of the population mean. Signup and view all the answers

When is the t-distribution used?

When the sample size is small and the population standard deviation is unknown Signup and view all the answers

Which of the following statements about the Central Limit Theorem is true?

The ideal sample size for applying it is at least 30. Signup and view all the answers

What is represented by $σ^2$ in the variance formula for the sampling distribution?

Population variance. Signup and view all the answers

What relationship exists between sampling error and sample size?

They are inversely related Signup and view all the answers

What does the correction factor in the standard deviation formula indicate?

It adjusts for sampling from finite populations Signup and view all the answers

As the sample size increases, the sampling distribution of the sample mean approaches which distribution?

Normal distribution Signup and view all the answers

In the context of confidence intervals, what does it mean if the interval may not contain the true parameter value?

There is uncertainty in the estimation of the population parameter Signup and view all the answers

What is the relationship between the sample mean ($ar{x}$) and population mean (μ) in the context of sampling distributions?

The sample mean can exceed the population mean. Signup and view all the answers

What is the degree of freedom used when calculating confidence intervals based on sample size?

n - 1 Signup and view all the answers

What is the main characteristic of the t-distribution compared to the normal distribution?

It has thicker tails than the normal distribution Signup and view all the answers

What does the standard deviation determine in the graph of the normal distribution?

The shape, height, and width of the curve Signup and view all the answers

Which symbol represents the mean of a discrete random variable?

μ Signup and view all the answers

What is the formula for calculating the standard deviation of a discrete random variable?

σ = √∑[X^2 P(X)] - μ^2 Signup and view all the answers

How does a change in the mean affect the graph of the normal curve?

It shifts the curve to the right or left Signup and view all the answers

What is indicated by the Empirical Rule (68-95-99.7)?

Percentages of data within standard deviations from the mean Signup and view all the answers

What is the relationship between the standard normal curve's mean and standard deviation?

μ = 0 and σ = 1 Signup and view all the answers

Why is the total area under the normal curve equal to 1?

Because it represents all possible outcomes Signup and view all the answers

Which component is NOT required to find the mean of a discrete random variable?

The standard deviation Signup and view all the answers

What does the notation < Θ < b indicate in estimation?

The estimated parameter is between two specific values. Signup and view all the answers

In the context of population proportions, what does p̂ (p-hat) represent?

The sample proportion derived from a sample. Signup and view all the answers

What is one key characteristic of the sampling distribution of the mean according to the Central Limit Theorem?

It approaches a normal distribution as sample size increases. Signup and view all the answers

How is the minimum sample size determined for estimating population proportions?

Using a specific formula based on desired confidence level. Signup and view all the answers

In a population of 1,000 people, if 346 have a rapid quarantine pass, what is the population proportion?

0.346 Signup and view all the answers

Why is it important to round up the sample size when conducting statistical analysis?

To increase the statistical reliability of the results. Signup and view all the answers

What is the relationship between sample size and the reliability of statistical inferences?

Larger sample sizes generally increase reliability. Signup and view all the answers

When discussing population parameters, which statement is true regarding the population proportion?

It is a fraction of the population with specific characteristics. Signup and view all the answers

Study Notes

Random Variables and Probability Distribution

A random variable associates a real number with each element in the sample space.
To determine random variables, establish the sample space and assign letters to outcomes.
Random variable values must have probabilities between 0 and 1 (0 ≤ P(X) ≤ 1).
The sum of probabilities for all values of a random variable should equal 1 (ΣP(X) = 1).

Mean and Variance of Discrete Random Variables

Mean is a central tendency measure, often called the average.
Mean (average) formula: Mean = Sum of scores / Total number of scores.
Mean of a discrete random variable is the sum of products of values and their probabilities.
Standard deviation indicates the spread of the distribution and is calculated as σ = √Σ[X²P(X)] - μ².

Normal Distribution Properties

The area under the normal curve equals 1 (100%).
The "Empirical Rule" (68-95-99.7) refers to the distribution of values in a normal curve.
Mean determines the position of the center of the curve; changes to the mean shift the graph.
Standard deviation affects graph shape, influencing the height and width of the curve.

Sampling Distribution of Sample Mean

The mean of sampling distribution of sample means equals the population mean (μX̅ = μ).
Variance of sampling distribution: σ²X̅ = σ²/n (for infinite population).
Standard deviation of sample means (standard error): σX̅ = σ / √n.
As sample size increases, the sampling distribution approaches normal distribution regardless of population shape.

T-Distribution

T-distribution resembles normal distribution but is used for small sample sizes (n < 30) with unknown population standard deviation.
Confidence intervals derive from sampling to estimate population parameters, specified as a < Θ < b.

Central Limit Theorem

As sample size (n) increases, the mean's sampling distribution approaches normality.
A population characteristic is represented as a fraction (p for population proportion and p̂ for sample proportion).

Sample Size Determination

Sample size (n) is crucial for making statistical inferences about the population.
A minimum sample size formula ensures adequate reliability when estimating population mean or proportion.

Example Calculations

For example, if 346 out of 1,000 people possess a rapid quarantine pass, the population proportion is calculated as 346/1000.

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Description

Test your understanding of random variables and probability distributions in this first quarter examination for Grade 11. This quiz covers essential concepts related to the behavior of random variables and their probability distributions.

Statistics & Probability Exam - Grade 11