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?

<p>It shows an approximately normal distribution for all possible sample sizes. (B)</p> 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)?

<p>$2$ (C)</p> Signup and view all the answers

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

<p>5.2 minutes (B)</p> Signup and view all the answers

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

<p>1.6 minutes (C)</p> Signup and view all the answers

How many customers were observed in the random sample?

<p>36 customers (B)</p> Signup and view all the answers

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

<p>$ rac{1.6}{36}$ (D)</p> Signup and view all the answers

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

<p>0.2667 minutes (B)</p> Signup and view all the answers

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

<p>0.5 (A)</p> Signup and view all the answers

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

<p>0.025 (A)</p> 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?

<p>0.9757 (A)</p> Signup and view all the answers

What best describes a sampling distribution of the sample means?

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

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

<p>Each sample mean represents a possible value in the sampling distribution. (C)</p> 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?

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

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

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

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

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

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

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

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

<p>0.5 (A)</p> Signup and view all the answers

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

<p>Convenience sampling (D)</p> 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?

<p>0.2535 (D)</p> Signup and view all the answers

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

<p>3 (B)</p> Signup and view all the answers

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

<p>1.97 (A)</p> 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?

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

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

<p>Normal distribution (B)</p> Signup and view all the answers

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

<p>35 (C)</p> Signup and view all the answers

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

<p>5 (A)</p> Signup and view all the answers

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

<p>Covers the lower 97.56% of data (B)</p> 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?

<p>0.6420 (C)</p> 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?

<p>-0.75 (D)</p> Signup and view all the answers

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

<p>The standard deviation of the sample means. (C)</p> 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?

<p>1.67 (D)</p> Signup and view all the answers

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

<p>$z = \frac{𝑥̄ - 𝜇}{𝜎 \cdot \sqrt{n}}$ (D)</p> 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?

<p>2.36 (D)</p> 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?

<p>0.5250 (B)</p> 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?

<p>1 (A)</p> Signup and view all the answers

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

<p>2.75 minutes (B)</p> Signup and view all the answers

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

<p>0.2 minutes (D)</p> Signup and view all the answers

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

<p>2.3 liters (D)</p> 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?

<p>Higher than 0.5 (C)</p> Signup and view all the answers

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

<p>7.0 years (B)</p> 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?

<p>₱400.00 (A)</p> 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?

<p>Greater than 0.6 (D)</p> 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?

<p>High probability (B)</p> Signup and view all the answers

Flashcards

Sampling Distribution of Sample Means

The probability distribution of the mean values obtained from a large number of samples from a population.

Mean of Sample Means (µ𝑥̄)

Equal to the population mean (µ).

Standard Error of the Mean (σ𝑥̄)

The standard deviation of the sample means, calculated as the population standard deviation (σ) divided by the square root of the sample size (n).

Central Limit Theorem

The theorem that states that the sampling distribution of the sample means will approach a normal distribution as the sample size (n) increases.

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Probability Histogram

A graph displaying the frequency or probability of different values in the sampling distribution (of sample means), indicating a normal distribution.

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Sampling Distribution

A distribution of all possible means from samples of a specific size taken from a particular population.

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Population Distribution

The distribution of all possible values of a variable within an entire population.

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Sampling Distribution of the Means

The probability distribution of sample means calculated from all possible samples of a specific size from a population.

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Sample Mean

The average value of a specific set of data from a sample.

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Standard Error

The standard deviation of a sampling distribution.

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Random Variable

A variable whose value is a numerical outcome of a random phenomenon.

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Sample Size

The number of data points in a sample.

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DepEd Order on "No periodical test"

A DepEd policy that eliminates periodical tests in schools to discourage cheating.

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What is the standard error of the mean?

The standard deviation of the sampling distribution of sample means. It measures how much the sample means vary around the population mean.

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Population Mean (µ)

The average value of a variable in the entire population.

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Sample Mean (𝑥ҧ)

The average value of a variable in a sample.

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Standard Deviation (𝜎)

A measure of how spread out the data is around the mean in the population.

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Sample Size (𝑛)

The number of individuals or items in a sample.

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Probability of a Sample Mean

The likelihood of obtaining a sample mean within a specific range.

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Z-Score for Sample Mean

A standardized score that measures how many standard errors a particular sample mean is away from the population mean.

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SND Table or Excel

Tools that provide probabilities associated with specific z-scores.

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Drive-Thru Time?

A food chain restaurant tracks drive-thru service times. They know the average time is 5.2 minutes, with a standard deviation of 1.6 minutes. If a random sample of 36 customers is observed, what's the probability their average time is between 5 and 5.5 minutes?

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Standard Error (SE)

The standard deviation of the sampling distribution of the sample means. It measures how much the sample means tend to vary from the population mean.

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Calculate SE

To calculate the Standard Error (SE), divide the population standard deviation (σ) by the square root of the sample size (n).

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What is the SE for this drive-thru?

In our example, the population standard deviation is 1.6 minutes, and the sample size is 36. Therefore, the SE is 1.6 / √36 = 0.2667 minutes.

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Probability Range

The probability of the sample mean falling between 5 and 5.5 minutes. The area under the normal distribution curve between those values represents this probability.

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Z-Score Calculation

A z-score is the number of standard deviations a value is away from the mean. It's calculated using the formula: z = (x - μ) / σx̅. This helps determine the probability.

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What is the z-score?

To calculate the z-score, we use the values x = 5 (lower limit), μ = 5.2 (population mean), and σx̅ = 0.2667 (SE). So, z = (5 - 5.2) / 0.2667 = -0.75.

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Probability Finding

Using a z-table or statistical software, find the probability associated with the calculated z-score. This will give us the probability of the sample mean falling within the desired range.

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Mean of Sample Means

The average of all possible sample means from a population. It's always equal to the population mean (μ).

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Standard Error of the Mean

The standard deviation of the sampling distribution of sample means. It measures the spread of sample means around the population mean. Calculated as (σ / √n), where σ is the population standard deviation and n is the sample size.

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What is the standard error of the mean for a sample of 25 from a population with a standard deviation of 5?

The standard error of the mean is calculated as σ / √n, so in this case it would be 5 / √25 = 1.

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What does a larger standard error of the mean indicate?

A larger standard error indicates that the sample means are more spread out around the population mean. This means that the sample mean is a less precise estimate of the population mean.

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How is the standard error of the mean affected by sample size?

The standard error of the mean is inversely proportional to the square root of the sample size. This means that as the sample size increases, the standard error decreases.

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What is the probability of obtaining a sample mean greater than 400.00 from a population with a mean of 380.00 and a standard deviation of 51.00, given a sample size of 75?

This problem requires using the z-score formula and standard normal distribution tables or software to calculate the probability. The z-score for a sample mean of 400.00 is (400 - 380) / (51 / √75) = 2.74. This means that the sample mean is 2.74 standard errors above the population mean. The probability of obtaining a sample mean greater than 400.00 is then 1 - P(z < 2.74), which can be found from a standard normal distribution table or using software.

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Sample Mean Duration

The average duration of a playlist consisting of 36 randomly selected songs from Kyle's music collection.

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Standard Error of the Sample Means

A measure of the variability of sample means around the population mean.

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Probability of Running Out of Water

The likelihood that the basketball team will use more than 45 liters of water during a game.

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Probability of Group Sleep

The likelihood that a random group of 35 public school students will get more than 6 hours of sleep per night.

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Probability of Smartphone Life

The likelihood that the average life of a random sample of 9 smartphones falls between 6.4 and 7.2 years.

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Required Sample Size

The number of samples (n) needed to ensure a 95% probability that the sample mean falls within a specific range.

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Probability of Blood Pressure

The likelihood that a single sample of 28 people will have an average systolic blood pressure less than 125.

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Probability of Water Usage

The likelihood that the average daily water usage of a random sample of 15 people falls between 120 and 125 liters.

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Probability of Teacher Salary

The likelihood that the average hourly salary of a sample of 75 online teachers is greater than ₱400.00.

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Study Notes

Important Note

  • This material is for personal use only
  • Respect the intellectual property rights of the author
  • Do not share or post the material on other platforms or social media accounts

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.

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