Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:
If you draw random samples of size n, then as n increases, the random variable ΣX consisting of sums tends to be normally distributed and ΣΧ ~ N((n)(μΧ), (
)(σΧ)).
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:
= standard deviation of
To find probabilities for sums on the calculator, follow these steps.
2nd DISTR
* * *
2:normalcdf
* * *
normalcdf
(lower value of the area, upper value of the area, (n)(mean), (
)(standard deviation))
where:
An unknown distribution has a mean of 90 and a standard deviation of 15. A sample of size 80 is drawn randomly from the population.* * *
Let X = one value from the original unknown population. The probability question asks you to find a probability for the sum (or total of) 80 values.
ΣX = the sum or total of 80 values. Since μX = 90, σX = 15, and n = 80,
~ N((80)(90), * * *
(
)(15))
(15)
a. Find P(Σx > 7,500)
P(Σx > 7,500) = 0.0127
{:}
normalcdf
(lower value, upper value, mean of sums, stdev
of sums)
The parameter list is abbreviated(lower, upper, (n)(μX,
(σX))
normalcdf
(7500,1E99,(80)(90),
(15)) = 0.0127
1E99 = 1099.
Press the EE
key for E.
b. Find Σx where z = 1.5.
Σx = (n)(μX) + (z)
(σΧ) = (80)(90) + (1.5)(
)(15) = 7,401.2
An unknown distribution has a mean of 45 and a standard deviation of eight. A sample size of 50 is drawn randomly from the population. Find the probability that the sum of the 50 values is more than 2,400.
To find percentiles for sums on the calculator, follow these steps.
2nd DIStR * * *
3:invNorm * * *
k = invNorm (area to the left of k, (n)(mean),
(standard deviation))
where:
In a recent study reported Oct. 29, 2012 on the Flurry Blog, the mean age of tablet users is 34 years. Suppose the standard deviation is 15 years. The sample of size is 50.
σx =
(15) = 106.07
The distribution is normal for sums by the central limit theorem.
normalcdf
(1,500, 1,800, (50)(34),
(15)) = 0.7974
k = invNorm
(0.80,(50)(34),
(15)) = 1,789.3
In a recent study reported Oct.29, 2012 on the Flurry Blog, the mean age of tablet users is 35 years. Suppose the standard deviation is ten years. The sample size is 39.
The mean number of minutes for app engagement by a tablet user is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of size 70.
=
(1) = 8.37 minutes
k = invNorm (0.95,(70)(8.2),
(1)) = 587.76 minutes
Ninety five percent of the sums of app engagement times are at most 587.76 minutes.
P(Σx ≥ 600) = normalcdf
(600,E99,(70)(8.2),
(1)) = 0.0009
The mean number of minutes for app engagement by a table use is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample size of 70.
Farago, Peter. “The Truth About Cats and Dogs: Smartphone vs Tablet Usage Differences.” The Flurry Blog, 2013. Posted October 29, 2012. Available online at http://blog.flurry.com (accessed May 17, 2013).
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed. Additionally, if the original population has a mean of μX and a standard deviation of σx, the mean of the sums is nμx and the standard deviation is
(σx) where n is the sample size.
The Central Limit Theorem for Sums: ∑X ~ N[(n)(μx),(
)(σx)]
Mean for Sums (∑X): (n)(μx)
The Central Limit Theorem for Sums z-score and standard deviation for sums:
Standard deviation for Sums (∑X):
(σx)
Use the following information to answer the next four exercises: An unknown distribution has a mean of 80 and a standard deviation of 12. A sample size of 95 is drawn randomly from the population.
Find the probability that the sum of the 95 values is greater than 7,650.
0.3345
Find the probability that the sum of the 95 values is less than 7,400.
Find the sum that is two standard deviations above the mean of the sums.
7,833.92
Find the sum that is 1.5 standard deviations below the mean of the sums.
Use the following information to answer the next five exercises: The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly.
Find the probability that the sum of the 40 values is greater than 7,500.
0.0089
Find the probability that the sum of the 40 values is less than 7,000.
Find the sum that is one standard deviation above the mean of the sums.
7,326.49
Find the sum that is 1.5 standard deviations below the mean of the sums.
Find the percentage of sums between 1.5 standard deviations below the mean of the sums and one standard deviation above the mean of the sums.
77.45%
Use the following information to answer the next six exercises: A researcher measures the amount of sugar in several cans of the same soda. The mean is 39.01 with a standard deviation of 0.5. The researcher randomly selects a sample of 100.
Find the probability that the sum of the 100 values is greater than 3,910.
Find the probability that the sum of the 100 values is less than 3,900.
0.4207
Find the sum with a z–score of –2.5.
3,888.5
Find the sum with a z–score of 0.5.
Find the probability that the sums will fall between the z-scores –2 and 1.
0.8186
Use the following information to answer the next four exercises: An unknown distribution has a mean 12 and a standard deviation of one. A sample size of 25 is taken. Let X = the object of interest.
What is the mean of ΣX?
What is the standard deviation of ΣX?
5
What is P(Σx = 290)?
What is P(Σx > 290)?
0.9772
True or False: only the sums of normal distributions are also normal distributions.
In order for the sums of a distribution to approach a normal distribution, what must be true?
The sample size, n, gets larger.
What three things must you know about a distribution to find the probability of sums?
An unknown distribution has a mean of 25 and a standard deviation of six. Let X = one object from this distribution. What is the sample size if the standard deviation of ΣX is 42?
49
An unknown distribution has a mean of 19 and a standard deviation of 20. Let X = the object of interest. What is the sample size if the mean of ΣX is 15,200?
Use the following information to answer the next three exercises.* A market researcher analyzes how many electronics devices customers buy in a single purchase. The distribution has a mean of three with a standard deviation of 0.7. She samples 400 customers.
What is the z-score for Σx = 840?
26.00
What is the z-score for Σx = 1,186?
What is P(Σx < 1,186)?
0.1587
Use the following information to answer the next three exercises:* An unkwon distribution has a mean of 100, a standard deviation of 100, and a sample size of 100. Let X = one object of interest.
What is the mean of ΣX?
What is the standard deviation of ΣX?
1,000
What is P(Σx > 9,000)?
Which of the following is NOT TRUE about the theoretical distribution of sums?
Suppose that the duration of a particular type of criminal trial is known to have a mean of 21 days and a standard deviation of seven days. We randomly sample nine trials.
Suppose that the weight of open boxes of cereal in a home with children is uniformly distributed from two to six pounds with a mean of four pounds and standard deviation of 1.1547. We randomly survey 64 homes with children.
Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,500. We randomly survey ten teachers from that district.
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