The Central Limit Theorem for Sums (Optional) Quiz
The question sheet
Reveal any answer as you study-
According to the CLT for sums, as sample size n increases, ΣX tends to be:
- uniformly distributed
- exponentially distributed
- normally distributed
- skewed right
Reveal answer
Answer: normally distributed
Source evidence
PDF page 399: If you draw random samples of size n, then as n increases, the random variable ΣX consisting of sums tends to be
PDF page 399: normally distributed and ΣΧ ~ N[(n)(μΧ), ( )(σΧ)].
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For the distribution of sums, the mean equals the original mean multiplied by:
- the standard deviation
- the square root of n
- one
- the sample size
Reveal answer
Answer: the sample size
Source evidence
PDF page 399: The central limit theorem for sums says that if you keep drawing larger and larger samples and taking their sums, the sums form their own normal distribution (the sampling distribution), which approaches a normal distribution as the sample size increases. The normal distribution has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size. The random variable ΣX has the following z-score associated with it:
PDF page 399: i. (n)(μX) = mean of ΣX
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For a population with μ=90 and n=80, the mean of the sums equals:
- 90
- 7,500
- 7,200
- 1,500
Reveal answer
Answer: 7,200
Source evidence
PDF page 399: • mean of the sums = (n)(μX) = (80)(90) = 7200
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For μ=90, σ=15, n=80, what is P(Σx > 7500)?
- 0.5000
- 0.0127
- 0.1500
- 0.9873
Reveal answer
Answer: 0.0127
Source evidence
PDF page 400: a. Find P(Σx > 7500)
PDF page 400: P(Σx > 7500) = 0.0127
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The sum of 80 values (Σx) in the mean-90 example is:
- 1,500
- 90
- 7,200
- 7,500
Reveal answer
Answer: 7,500
Source evidence
PDF page 400: • sum of 80 values = Σx = 7500
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For tablet users with μ=34 and n=50, the mean of the sum of ages is:
- 1,800
- 34
- 1,700
- 1,500
Reveal answer
Answer: 1,700
Source evidence
PDF page 401: a. μΣx = nμx = 50(34) = 1,700 and σΣx =
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The distribution of the sum of tablet users' ages is:
- uniform
- skewed
- binomial
- normal
Reveal answer
Answer: normal
Source evidence
PDF page 401: The distribution is normal for sums by the central limit theorem.
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The 80th percentile for the sum of 50 tablet ages equals about:
- 1,500
- 1,700
- 1,789.3
- 34
Reveal answer
Answer: 1,789.3
Source evidence
PDF page 401: k = invNorm(0.80,(50)(34), (15)) = 1789.3
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For app engagement with μ=8.2 and n=70, the mean of the sums is:
- 70 minutes
- 574 minutes
- 8.2 minutes
- 600 minutes
Reveal answer
Answer: 574 minutes
Source evidence
PDF page 401: a. μΣx = nμx = 70(8.2) = 574 minutes and σΣx =
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The 95th percentile for the sum of 70 app engagement times is about:
- 600 minutes
- 574 minutes
- 8.2 minutes
- 587.76 minutes
Reveal answer
Answer: 587.76 minutes
Source evidence
PDF page 401: k = invNorm (0.95,(70)(8.2), (1)) = 587.76 minutes Ninety-five percent of the app engagement times are at most 587.76 minutes.
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In the app engagement example, 10 hours equals how many minutes?
- 600 minutes
- 574 minutes
- 587.76 minutes
- 100 minutes
Reveal answer
Answer: 600 minutes
Source evidence
PDF page 401: c. 10 hours = 600 minutes
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To find probabilities for sums on the TI calculator, you use which command?
- binompdf
- invNorm
- geometpdf
- normalcdf
Reveal answer
Answer: normalcdf
Source evidence
PDF page 399: To find probabilities for sums on the calculator, follow these steps: nd
PDF page 399: 2:normalcdf normalcdf(lower value of the area, upper value of the area, (n)(mean), ( )(standard deviation))
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