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The standard error of the sampling distribution is always greater than the population standard deviation.
False
The mean of the sampling distribution of the mean is different from the population mean.
False
A sample size of 25 is generally considered large enough for the Central Limit Theorem to apply.
False
As the sample size increases, the variance of the sampling distribution of the mean decreases.
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When the population distribution is not normal, the sampling distribution of the mean will still be approximately normal for larger sample sizes.
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The sampling distribution of the mean can be viewed as symmetric bell-shaped regardless of the sample size.
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The mean of the sampling distribution is represented as σx̄.
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The larger the sample size, the closer the sampling distribution resembles a normal distribution.
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The sampling distribution is a theoretical probability distribution of a sample statistic.
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The sample mean is a fixed value and does not vary among different samples.
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Larger sample sizes usually provide more accurate estimates of population parameters.
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The standard deviation of the sampling distribution of the mean is equal to the population standard deviation.
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The population mean can be estimated accurately without considering the sampling distribution.
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The variance of the sampling distribution of the mean is calculated by the square root of the standard deviation of the population.
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Calculating the sampling distribution involves drawing all possible distinct samples from the population.
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The standard error of the mean is calculated as $0.4$ using the formula $ rac{2}{ ext{square root of } 25}$.
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A population of individuals must consist of at least 10 members to perform a sampling distribution analysis.
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The average monthly salary of graphic designers is $15,500$ with a probability of $0.3632$ that it exceeds $16,200$.
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If 64 graphic designers are randomly selected, the probability that their average monthly salary exceeds $16,200$ is $0.9474$.
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The Central Limit Theorem can be applied if the sample size is sufficiently large, regardless of the population distribution shape.
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The formula for population proportion is $p = rac{x}{N}$ where $x$ is the number of successes in the population.
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The mean of the sampling distribution of the sample proportion is different from the population proportion.
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The probability of the average monthly salary of five randomly selected graphic designers exceeding $16,200$ is $0.2177$.
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A sample proportion is calculated by multiplying the number of successes by the sample size.
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The standard error of the sample proportion is calculated using the formula σp̂ = √(p(1-p)/n).
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The sampling distribution of the sample proportion can be approximated by a normal distribution only when n(1-p) ≥ 10.
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In a random sample of 50 university students, if the true proportion p is 0.4, then the standard error of the proportion is approximately 0.0693.
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The mean of the sampling distribution of the difference between two sample means is given by μ(X̄ - Ȳ) = μx + μy.
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The probability that more than half of the sampled students made an in-app purchase last month is found to be 0.0749.
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If X and Y are normally distributed, then their difference X - Y will not necessarily be normally distributed.
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The variance of the sampling distribution of the difference between two sample means is calculated using the formula σ²(X̄ - Ȳ) = σ²x / nx + σ²y / ny.
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For a sampling distribution of the sample proportion, it is not necessary for both np and n(1-p) to be at least 5.
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Study Notes
Sampling Distribution
- A theoretical probability distribution of a sample statistic.
- Sample statistic is a random variable, such as the sample mean, sample proportion, etc.
- Calculated by drawing all possible samples of the same size from a population.
Sampling Distribution of the Mean
- For statistical inferences, draw a sample from a population to estimate the population mean (μ) using the sample mean (X̄).
- X̄ varies among samples making it a random variable.
- Its probability distribution is called the sampling distribution of the mean.
Calculations for Sampling Distribution of the Mean
- Can be calculated by drawing all possible distinct samples from the population.
- If there are m distinct samples, the possible values of X̄ are X1, X2, X3,..., Xm, forming a population of sample mean values.
- Population mean (μx̄) is calculated by adding all the sample means and dividing by the number of samples (m).
- The standard deviation (σx̄) of the sampling distribution is calculated by taking the square root of the variance (σ²x̄).
Sampling Distribution of the Sample Proportion
- Represents the proportion of the population that possesses a particular characteristic of interest.
- For a population of size N, if the number of males is x, the proportion of males is denoted by p and is given by: p = x/N
Sample Proportion (p̂)
- The sample proportion provides an estimate of the population proportion.
- Calculated by dividing the number of successes (x) in a sample by the sample size (n).
- The mean (μp̂) of the sampling distribution of the sample proportion is equal to the population proportion.
Standard Error of the Sample Proportion
- The standard error (σp̂) of the sampling distribution of the sample proportion is σp̂ = √(p(1-p)/n)
Sampling Distribution of the Difference Between Two Sample Means
- The need often arises to compare the means between two independent populations by studying the difference between two relevant sample means.
- Two samples are independent if the selection of items in one sample has no effect on the selection of items in the other sample.
- The mean of the sampling distribution of the difference between two sample means is: μ(X̄ - Ȳ ) = μx - μy
- The variance of the sampling distribution of the difference between two sample means is: σ²(X̄ - Ȳ ) = σ²x / nx + σ²y / ny
- If X and Y are normally distributed, then X - Y will also be normally distributed with the mean and variance as stated above.
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Description
This quiz covers the concept of sampling distribution, focusing on the sampling distribution of the mean. Understand how sample statistics act as random variables and explore the calculations involved in estimating population means using sample means. Test your knowledge on this fundamental statistical concept.
