Asked what the central limit theorem says, a student replies, As you take larger
ID: 3310395 • Letter: A
Question
Asked what the central limit theorem says, a student replies, As you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal.
Is the student right?
No. As you take larger and larger samples the histogram of the sample values looks less Normal.
To estimate the mean height of male students on your campus, you will measure an SRS of students. You know from government data that the standard deviation of the heights of young men is about 3.3 inches. You want your sample mean x¯¯¯x¯ to estimate with an error of no more than one-half inch in either direction.
What standard deviation (±0.0001) must x¯¯¯x¯ have so that 99.7% of all samples give an x¯¯¯x¯ within one-half inch of ? (Use the 68-95-99.7 rule)
How large an SRS do you need to reduce the standard deviation of x¯¯¯x¯ to the value you found in the previous part?
Asked what the central limit theorem says, a student replies, As you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal.
Is the student right?
No. The central limit theorem says nothing about the histogram of the sample values. It deals only with the distribution of the sample means. Yes. This is exactly what the theorem says.No. As you take larger and larger samples the histogram of the sample values looks less Normal.
To estimate the mean height of male students on your campus, you will measure an SRS of students. You know from government data that the standard deviation of the heights of young men is about 3.3 inches. You want your sample mean x¯¯¯x¯ to estimate with an error of no more than one-half inch in either direction.
What standard deviation (±0.0001) must x¯¯¯x¯ have so that 99.7% of all samples give an x¯¯¯x¯ within one-half inch of ? (Use the 68-95-99.7 rule)
How large an SRS do you need to reduce the standard deviation of x¯¯¯x¯ to the value you found in the previous part?
Explanation / Answer
The Central Limit Theorem states that the sampling distribution of the sampling means approaches a normal distribution as the sample size gets larger — no matter what the shape of the population distribution. This fact holds especially true for sample sizes over 30. All this is saying is that as you take more samples, especially large ones, your graph of the sample means will look more like a normal distribution.
Hence answer is
No. The central limit theorem says nothing about the histogram of the sample values. It deals only with the distribution of the sample means.Related Questions
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