Suppose we take repeated random samples of size 20 from a population with a mean of 60 and a standard deviation of 8. Which of the following statements is true about the sampling distribution of the sample mean (x̄)? Check all that apply. The distribution is normal regardless of the shape of the population distribution, because the sample size is large enough. The distribution will be normal as long as the population distribution is normal. The distribution's mean is the same as the population mean 60. The distribution's standard deviation is larger than the population standard deviation of 8.
Question
Suppose we take repeated random samples of size 20 from a population with a mean of 60 and a standard deviation of 8. Which of the following statements is true about the sampling distribution of the sample mean (x̄)? Check all that apply. The distribution is normal regardless of the shape of the population distribution, because the sample size is large enough. The distribution will be normal as long as the population distribution is normal. The distribution's mean is the same as the population mean 60. The distribution's standard deviation is larger than the population standard deviation of 8.
Solution 1
The correct statements are:
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The distribution's mean is the same as the population mean 60. This is a property of sampling distributions: the mean of the sampling distribution (also known as the expected value of the sample mean) is equal to the population mean.
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The distribution will be normal as long as the population distribution is normal. This is part of the Central Limit Theorem, which states that if the population from which we are sampling is normally distributed, then the sampling distribution will also be normal, regardless of sample size.
However, the other two statements are incorrect:
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The distribution is normal regardless of the shape of the population distribution, because the sample size is large enough. This is a common misunderstanding of the Central Limit Theorem. The theorem states that if we take a large enough sample (usually n > 30 is considered "large enough"), the sampling distribution will be approximately normal, regardless of the shape of the population distribution. However, in this case, the sample size is 20, which is not large enough to guarantee a normal distribution.
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The distribution's standard deviation is larger than the population standard deviation of 8. This is incorrect. The standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size. So in this case, the standard deviation of the sampling distribution would be 8/√20, which is less than 8.
Solution 2
The correct statements are:
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The distribution's mean is the same as the population mean 60. This is a property of sampling distributions: the mean of the sampling distribution (also known as the expected value of the sample mean) is equal to the population mean.
-
The distribution will be normal as long as the population distribution is normal. This is part of the Central Limit Theorem, which states that if the population from which we are sampling is normally distributed, then the sampling distribution will also be normal, regardless of sample size.
The incorrect statements are:
-
The distribution is normal regardless of the shape of the population distribution, because the sample size is large enough. This is a common misunderstanding of the Central Limit Theorem. The theorem states that if the sample size is "large enough" (usually considered to be greater than or equal to 30), the sampling distribution will be approximately normal, regardless of the shape of the population distribution. However, the sample size here is 20, which is not considered "large enough" for the theorem to apply.
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The distribution's standard deviation is larger than the population standard deviation of 8. This is incorrect. The standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size. So in this case, the standard error would be 8/sqrt(20), which is less than 8.
Solution 3
The correct statements are:
-
The distribution's mean is the same as the population mean 60. This is a property of sampling distributions - the mean of the sampling distribution (also known as the expected value of the sample mean) is equal to the population mean.
-
The distribution will be normal as long as the population distribution is normal. This is part of the Central Limit Theorem, which states that if the population from which we are sampling is normally distributed, then the sampling distribution will also be normal, regardless of sample size.
The incorrect statements are:
-
The distribution is normal regardless of the shape of the population distribution, because the sample size is large enough. This is a common misunderstanding of the Central Limit Theorem. The theorem states that the sampling distribution will approach a normal distribution as the sample size increases, but it does not guarantee normality for any specific sample size. In practice, a sample size of 20 is often not large enough to ensure a normal distribution unless the population distribution is already approximately normal.
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The distribution's standard deviation is larger than the population standard deviation of 8. This is incorrect. The standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size. So, for a sample size of 20 and a population standard deviation of 8, the standard error would be 8/√20 = 1.79, which is smaller than the population standard deviation.
Similar Questions
Which of the following statements about the sampling distribution of the sample mean, x-bar, is true? Check all that apply. The distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is large enough. The distribution is normal regardless of the sample size, as long as the population distribution is normal. The distribution's mean is the same as the population mean. The distribution's standard deviation is smaller than the population standard deviation.
5. Suppose the random variable X𝑋 is normally distributed with μ=18𝜇=18 and σ2=64𝜎2=64 . If the sample size is equal to 20, what is the sampling distribution of the sample mean, X¯𝑋¯ ?Multiple choice 3 Question 3 N(20,1.79)𝑁(20,1.79) N(18,3.2)𝑁(18,3.2) N(18,1.79)𝑁(18,1.79) N(18,64)𝑁(18,64)
Sampling Distributions Checkpoint 2Question 1Select all that apply.10 pointsWhich of the following statements about the sampling distribution of the sample mean, x-bar, is true? Check all that apply. The distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is large enough. The distribution is normal regardless of the sample size, as long as the population distribution is normal. The distribution's mean is the same as the population mean. The distribution's standard deviation is smaller than the population standard deviation.Question 2Select one answer.10 pointsPictured below (in scrambled order) are three histograms. One of them represents a population distribution. The other two are sampling distributions of x-bar: one for sample size n = 5 and one for sample size n = 40.Based on the histograms, what is the most likely value of the population mean? 290 1 8 5Question 3Select one answer.10 pointsSuppose that a candy company makes a candy bar whose weight is supposed to be 50 grams, but in fact, the weight varies from bar to bar according to a normal distribution with mean μ = 50 grams and standard deviation σ = 2 grams.If the company sells the candy bars in packs of 4 bars, what can we say about the likelihood that the average weight of the bars in a randomly selected pack is 4 or more grams lighter than advertised? There is no way to evaluate this likelihood, since the sample size (n = 4) is too small. There is about a 16% chance of this occurring. There is about a 2.5% chance of this occurring. It is extremely unlikely for this to occur; the probability is very close to 0. There is about a 5% chance of this occurring.Question 4Select one answer.10 pointsWhen the population is not normally distributed, the sampling distribution of the mean approximates which of the following? A distribution that is not normal A slight positive skew A normal distribution given a large enough sample A normal distribution
The sampling distribution of sample mean for a large population is approximately normal if the sample size is
A distribution of a single statistic from repeated random samples of the same size from the same population refers to which of the following? Distribution of summary statistics The normal curve Distribution of population parameters Sampling distribution of a statistic Random sampling
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