Podcast
Questions and Answers
What key condition must be met for the Central Limit Theorem to be applicable?
What key condition must be met for the Central Limit Theorem to be applicable?
- The sample size must be sufficiently large. (correct)
- The samples must be drawn from a finite population.
- The population standard deviation must be known.
- The population must be normally distributed.
According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean as the sample size increases?
According to the Central Limit Theorem, what happens to the sampling distribution of the sample mean as the sample size increases?
- It approaches a normal distribution. (correct)
- It becomes more skewed.
- It approaches the population distribution.
- It becomes uniform.
A population has a mean of $\mu$ and a standard deviation of $\sigma$. According to the Central Limit Theorem, what are the mean ($\mu_{\bar{x}}$) and standard deviation ($\sigma_{\bar{x}}$) of the sampling distribution of the sample mean for samples of size n?
A population has a mean of $\mu$ and a standard deviation of $\sigma$. According to the Central Limit Theorem, what are the mean ($\mu_{\bar{x}}$) and standard deviation ($\sigma_{\bar{x}}$) of the sampling distribution of the sample mean for samples of size n?
- $\mu_{\bar{x}} = n\mu$, $\sigma_{\bar{x}} = n\sigma$
- $\mu_{\bar{x}} = \mu$, $\sigma_{\bar{x}} = \sigma/\sqrt{n}$ (correct)
- $\mu_{\bar{x}} = \mu$, $\sigma_{\bar{x}} = \sigma$
- $\mu_{\bar{x}} = \mu/n$, $\sigma_{\bar{x}} = \sigma/n$
A researcher wants to estimate the average height of adults in a city. They plan to take multiple random samples and calculate the mean height for each sample. How does the Central Limit Theorem help in this scenario?
A researcher wants to estimate the average height of adults in a city. They plan to take multiple random samples and calculate the mean height for each sample. How does the Central Limit Theorem help in this scenario?
What is the purpose of converting sample mean values to z-values when solving problems related to the Central Limit Theorem?
What is the purpose of converting sample mean values to z-values when solving problems related to the Central Limit Theorem?
A factory produces light bulbs with an average lifespan of 1000 hours and a standard deviation of 100 hours. If a random sample of 25 bulbs is selected, what is the standard deviation of the sampling distribution of the sample mean?
A factory produces light bulbs with an average lifespan of 1000 hours and a standard deviation of 100 hours. If a random sample of 25 bulbs is selected, what is the standard deviation of the sampling distribution of the sample mean?
A population has a non-normal distribution with a mean of 50 and a standard deviation of 15. If we take a large sample (n > 30), what can we say about the sampling distribution of the sample mean?
A population has a non-normal distribution with a mean of 50 and a standard deviation of 15. If we take a large sample (n > 30), what can we say about the sampling distribution of the sample mean?
The average score on a standardized test is 70 with a standard deviation of 10. If 100 students are randomly selected, what is the probability that their average score will be greater than 72? (Assume the sampling distribution is approximately normal)
The average score on a standardized test is 70 with a standard deviation of 10. If 100 students are randomly selected, what is the probability that their average score will be greater than 72? (Assume the sampling distribution is approximately normal)
A car company claims that its new model gets an average of 35 miles per gallon (mpg) with a standard deviation of 3 mpg. A consumer group tests 49 cars and finds the sample mean to be 34 mpg. What is the z-score for this sample mean?
A car company claims that its new model gets an average of 35 miles per gallon (mpg) with a standard deviation of 3 mpg. A consumer group tests 49 cars and finds the sample mean to be 34 mpg. What is the z-score for this sample mean?
The weights of bags of sugar are normally distributed with a mean of 5 lbs and a standard deviation of 0.25 lbs. If you select a sample of 16 bags, what is the probability that the mean weight of the sample will be less than 4.9 lbs?
The weights of bags of sugar are normally distributed with a mean of 5 lbs and a standard deviation of 0.25 lbs. If you select a sample of 16 bags, what is the probability that the mean weight of the sample will be less than 4.9 lbs?
A certain airline claims that its flights are, on average, 90% on time. Suppose you sample 36 flights and find that only 80% were on time. Assuming the airline's claim is true, what is the probability of observing a sample proportion as low as 80%?
A certain airline claims that its flights are, on average, 90% on time. Suppose you sample 36 flights and find that only 80% were on time. Assuming the airline's claim is true, what is the probability of observing a sample proportion as low as 80%?
A researcher is studying the average number of hours that students study per week. The population mean is 15 hours, and the standard deviation is 4 hours. If the researcher takes a sample of 64 students, what is the range that captures the middle 95% of the sampling distribution of the sample mean?
A researcher is studying the average number of hours that students study per week. The population mean is 15 hours, and the standard deviation is 4 hours. If the researcher takes a sample of 64 students, what is the range that captures the middle 95% of the sampling distribution of the sample mean?
In a large city, the average commute time is 30 minutes with a standard deviation of 5 minutes. If we randomly select 100 commuters, what is the probability that their average commute time is between 29 and 31 minutes? (Assume the sampling distribution is approximately normal.)
In a large city, the average commute time is 30 minutes with a standard deviation of 5 minutes. If we randomly select 100 commuters, what is the probability that their average commute time is between 29 and 31 minutes? (Assume the sampling distribution is approximately normal.)
A pharmaceutical company is testing a new drug to lower blood pressure. The average blood pressure in the population is 130 mmHg with a standard deviation of 15 mmHg. If the company tests the drug on a sample of 25 patients, what is the probability that the sample mean blood pressure will be less than 125 mmHg? (Assume the sampling distribution is approximately normal.)
A pharmaceutical company is testing a new drug to lower blood pressure. The average blood pressure in the population is 130 mmHg with a standard deviation of 15 mmHg. If the company tests the drug on a sample of 25 patients, what is the probability that the sample mean blood pressure will be less than 125 mmHg? (Assume the sampling distribution is approximately normal.)
In a certain city, the average rent for a one-bedroom apartment is $1200 with a standard deviation of $200. A real estate company samples 64 apartments. What is the probability that the sample mean rent will be greater than $1250? (Assume the sampling distribution is approximately normal)
In a certain city, the average rent for a one-bedroom apartment is $1200 with a standard deviation of $200. A real estate company samples 64 apartments. What is the probability that the sample mean rent will be greater than $1250? (Assume the sampling distribution is approximately normal)
Flashcards
Central Limit Theorem
Central Limit Theorem
The theorem states that the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.
Central Limit Theorem Definition
Central Limit Theorem Definition
If random samples of size n is drawn from a population, then as n becomes larger, the sampling distribution of the mean approaches the normal distribution, regardless of the shape of the population
Central Limit Theorem Properties
Central Limit Theorem Properties
When all possible samples of size 'n' are drawn from a population, the frequency distribution of the sample means has specific properties.
Mean of Sample Means
Mean of Sample Means
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Standard Deviation of Sample Means
Standard Deviation of Sample Means
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Normality of Sample Means
Normality of Sample Means
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Steps to Solving Probabilities
Steps to Solving Probabilities
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What is probability (a)?
What is probability (a)?
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What is probability (b)?
What is probability (b)?
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Study Notes
- At the end of this lesson, you should be able to:
- Illustrate the Central Limit Theorem
- Define the sampling distribution of the sample mean using the Central Limit Theorem
- Solve problems involving sampling distributions of the sample mean
The Central Limit Theorem
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If random samples of size n are drawn from a population, then as n becomes larger, the sampling distribution of the mean approaches the normal distribution, regardless of the shape of the original population distribution.
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If all possible sample size n are drawn from a population with mean μ and standard deviation σ, and the sample mean x is calculated from each sample; then, the frequency distribution of x has these properties:
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Its mean is equal to the population mean (μx = μ)
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Its standard deviation is equal to the standard deviation of the population divided by the square root of the sample size (σx = σ / √n)
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It will tend to have a normal distribution, regardless of the shape of the population.
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In symbol form is expressed as: z = (x - μ) / (σ / √n)
Solving for Probabilities for Sample Mean
- To solve for probabilities for the Sample Mean:
- Find the value of the population mean
- Identify the event of interest in terms of sample mean (x) and find the appropriate area on the normal curve
- Convert all values of x to z-values using z = (x - μ) / (σ / √n)
- Use the Standard Normal Distribution Table (z-table) to compute the probability
Example Problems
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College students take 46.2 minutes to complete a certain exam on average, with a standard deviation of 8 minutes and normally distributed variable
- The probability a randomly selected college student will complete the exam in less than 43 minutes.
- How likely is it that a group of 50 students will take, on average, less than 43 minutes to complete the test
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A cup of a certain brand of ice cream has 660mg of cholesterol on average, and a standard deviation of 35mg with a variable that is normally distributed
- What is the probability that a cup of ice cream will have more than 670 mg of cholesterol
- How likely is it that a sample of 10 cups will have an average of larger than 670mg
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An investigator of a case of food poisoning found that the amount of salmonella in every serving of food is normally distributed with an average of 3.7 colony forming units per grams (cfu/g) and a standard deviation of 1.19 cfu/g
- Compute the probability that a serving has at least 4.2 cfu/g of salmonella
- Determine the probability that a sample of 10 servings has at least 4.2 cfu/g of salmonella
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Shinwa Company says its batteries last for 8 hours when used in DC motors, with a standard deviation of 3.8 hours, which is approximately normally distributed
- Find the probability that a sample of 10 batteries will have a mean lifespan of less than 6 hours in DC Motors
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A TV commercial aired a total of 30 times with sales of the advertised product averaging Php 100,000 a week with a standard deviation of Php 75,000
- Determine the probability that the sample mean belongs to the interval Php 90,000 to Php 110,000
- Find the interval if the mean encompasses 95% of the distribution of the sample mean
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