Statistics Chapter 3: Sampling Methods

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Questions and Answers

What is the probability associated with a sample mean of 1/3 in the sampling distribution?

0.375 (correct)

0.625

0.5

0.125

According to the properties of the distribution of sample means, what happens when the sample size (n) is large?

The sample means will remain the same as the population mean.

The standard error increases significantly.

The population mean differs from the sample mean.

The sample means will be normally distributed regardless of the population distribution. (correct)

What is the standard error (SE) of the sample means represented as?

$rac{ ext{Sample Mean}}{ ext{Population Size}}$

$ ext{Population Standard Deviation} imes ext{Sample Size}$

$rac{ ext{Population Standard Deviation}}{ ext{Sample Size}}$ (correct)

$rac{ ext{Population Mean}}{ ext{Sample Size}}$

What does the probability histogram of the sampling distribution indicate about repeated sampling?

It shows an approximately normal distribution for all possible sample sizes. (B) Signup and view all the answers

If the population standard deviation is $10$ and the sample size is $25$, what is the standard error (SE)?

$2$ (C) Signup and view all the answers

What is the mean time spent by customers at the drive-thru according to the provided data?

5.2 minutes (B) Signup and view all the answers

What is the standard deviation of the time spent by customers at the drive-thru?

1.6 minutes (C) Signup and view all the answers

How many customers were observed in the random sample?

36 customers (B) Signup and view all the answers

What is the formula to compute the standard error (SE) based on the provided data?

$rac{1.6}{36}$ (D) Signup and view all the answers

What is the standard error (SE) of the time spent at the drive-thru window?

0.2667 minutes (B) Signup and view all the answers

Using the z-score formula, what z-score corresponds to a mean time of 5.5 minutes?

0.5 (A) Signup and view all the answers

What is the probability that the mean time for the customers is below 5 minutes?

0.025 (A) Signup and view all the answers

What is the probability that the mean time at the drive-thru window is between 5 to 5.5 minutes?

0.9757 (A) Signup and view all the answers

What best describes a sampling distribution of the sample means?

A distribution of all possible means from multiple samples of a specific size. (A) Signup and view all the answers

Which of the following statements is true about the means calculated from a sample?

Each sample mean represents a possible value in the sampling distribution. (C) Signup and view all the answers

In the provided sample data of responses from SHS students, what value represents a sample mean favoring the DepEd Order?

2/3 (B) Signup and view all the answers

What is meant by 'standard error' in the context of sampling distribution?

The measure of how much the sample mean is expected to vary from the population mean. (C) Signup and view all the answers

Which option describes the effect of increasing the sample size on the standard error?

Standard error will decrease as sample size increases. (D) Signup and view all the answers

Why is the area under the normal curve important in statistics?

It provides a way to calculate probabilities for different outcomes. (B) Signup and view all the answers

When analyzing a sample like (1,1,0), what is the calculated mean?

0.5 (A) Signup and view all the answers

Which sampling method might lead to biased results if not applied correctly?

Convenience sampling (D) Signup and view all the answers

What is the standard error computed for the sample of 35 students with a population standard deviation of 1.5?

0.2535 (D) Signup and view all the answers

What is the population mean ($\mu$) in the study about the level of happiness among SHS students?

3 (B) Signup and view all the answers

If the sample mean ($\overline{x}$) is 3.5, what is the corresponding z-score for this value?

1.97 (A) Signup and view all the answers

In the context of this study, if $P(\overline{x} < 3.5) = 0.9756$, what does this imply about the sample mean of happiness?

There is a 97.56% chance the mean happiness is lower than 3.5. (A) Signup and view all the answers

What shape does the distribution of sample means follow according to the Central Limit Theorem?

Normal distribution (B) Signup and view all the answers

Using the variables provided, what is the sample size (n) for the study?

35 (C) Signup and view all the answers

Which of the following values indicates the highest level of happiness in the study among SHS students?

5 (A) Signup and view all the answers

In a normal distribution, where does a z-score of 1.97 fall in terms of probability?

Covers the lower 97.56% of data (B) Signup and view all the answers

What is the probability that the average time spent in the drive-thru for a sample of 36 customers is between 5 and 5.5 minutes?

0.6420 (C) Signup and view all the answers

If the sample mean 𝑥̄ = 5 and 𝜇𝑥̄ = 5.2, what is the value of z when 𝑥̄ = 5?

-0.75 (D) Signup and view all the answers

What does the standard error represent in the context of sampling distributions?

The standard deviation of the sample means. (C) Signup and view all the answers

For a sample size of 36, population mean of 55, and population standard deviation of 10, what is the standard error?

1.67 (D) Signup and view all the answers

What is the formula used for calculating the z-score of a sample mean?

$z = \frac{𝑥̄ - 𝜇}{𝜎 \cdot \sqrt{n}}$ (D) Signup and view all the answers

If the national average percentage score for a subject is 36.66 with a standard deviation of 3.57, what z-score corresponds to a sample average of 45% from a sample of 200?

2.36 (D) Signup and view all the answers

For a sample of 75 online teachers with a mean salary of ₱380.00 and a standard deviation of ₱51.00, what is the probability that the sample mean is less than ₱400.00?

0.5250 (B) Signup and view all the answers

What is the variance of the sampling distribution for a sample size of 16 with a population standard deviation of 4?

1 (A) Signup and view all the answers

What is the mean duration for a sample of 36 songs chosen from Kyle's collection?

2.75 minutes (B) Signup and view all the answers

What is the standard error of the sample means for Kyle's song playlist?

0.2 minutes (D) Signup and view all the answers

What is the average number of liters of water consumed by each basketball player during a game?

2.3 liters (D) Signup and view all the answers

What is the expected probability that a randomly selected group of 35 public school students gets more than 6 hours of sleep?

Higher than 0.5 (C) Signup and view all the answers

What is the mean life of a smartphone based on the given statistics?

7.0 years (B) Signup and view all the answers

If the average online teacher's salary is ₱380.00, what sample mean would indicate a significant probability of being greater than this amount?

₱400.00 (A) Signup and view all the answers

What is the probability that a random sample of 15 people uses between 120 and 125 liters of water daily?

Greater than 0.6 (D) Signup and view all the answers

In the scenario presented, if systolic blood pressures are normally distributed, what is the probability that a sample will have a mean blood pressure of less than 125?

High probability (B) Signup and view all the answers

Study Notes

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Chapter 3: Sampling Methods and Sampling Distribution

Prepared by: Grace Aquino Pasion
Date: October 2024

Module 2: Sampling Distribution of the Sample Means

Reference:
- Francisco, J., Laviña, C., Pasion, G. (2023). Statistics and Probability (A Modern Approach for the New Generation). Johnny and Hansel Publications, Quezon City

Module Topics

Sampling Distribution
Population Distribution
Sampling Distribution of the Means
Standard Error
Area under the normal curve
Problem Solving

Definition of Sampling Distribution of the Sample Means

A distribution of all possible means of a given sample size drawn from a specific population

Example 1

Entrance test results for incoming Grade 11 students in a certain school has a mean of 84 and a standard deviation of 8.8
Test scores follow a normal distribution
What is the probability that the mean test score of 25 examinees is above 85?

Example 2

A study about the level of happiness is done among SHS students.
Scores follow a normal distribution of 1 to 5.
The highest score is 5
Population mean: 3
Population standard deviation: 1.5
What is the probability that the mean level of happiness of 35 students is lower than 3.5?

Example 3

A food chain restaurant uses a drive-thru speed-of-service timer.
Average normal time is 5.2 minutes.
Standard deviation is 1.6 minutes
Sample size: 36 customers
Find the probability that the mean time at the drive-thru window is between 5 to 5.5 minutes.

Exercises

Compute the mean, variance, and standard error of the sampling distribution
Population mean: μ
Population standard deviation: σ
Sample size: n
Given values for n, μ, and σ for problem examples.

Module 2 Sampling Distribution of the Sample Means

Exercises from textbook
Average online teachers' salary and standard deviation
National average percentage score of Grade 6 in NAT
The number of hours spent playing mobile legends

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Description

This quiz covers Chapter 3 on Sampling Methods and Sampling Distribution from the textbook 'Statistics and Probability (A Modern Approach for the New Generation)'. Explore key concepts like sampling distribution, standard error, and area under the normal curve. Test your understanding of these topics and how they apply to population data.