Distributions of Sums of Samples

Learning Outcomes

  • Calculate probabilities for the sampling distribution for a sum using technology
  • Calculate percentiles for the sampling distribution for a sum using technology

Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution) and suppose:

  1. μX = the mean of Χ
  2. σΧ = the standard deviation of X

If you draw random samples of size n, then as n increases, the random variable [latex]\sum X[/latex] consisting of sums tends to be normally distributed and

[latex]\displaystyle {\sum{X}{\sim}{N}((n)(\mu_X), (\sqrt{n})(\sigma_X))}[/latex].

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:

  1. [latex]\sum x[/latex] is one sum.
  2. [latex]{z}=\frac{{\sum{x}-{({n})}{({\mu}_{{X}})}}}{{{(\sqrt{{n}})}{({\mu}_{{X}})}}}[/latex]
    1. [latex]{({n})}{({\mu}_{{X}})}=\text{the mean of }\sum{X}[/latex]
    2. [latex]{\left(\sqrt{{n}}\right)}{\left({\sigma}_{{X}}\right)} =\text{the standard deviation of }\sum{X}[/latex]

USING THE TI-83, 83+, 84, 84+ CALCULATOR

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), ([latex]\displaystyle\sqrt{{n}}[/latex])(standard deviation))

where:

  • mean is the mean of the original distribution,
  • standard deviation is the standard deviation of the original distribution, and
  • sample size = n

Recall: Scientific Notation

Scientific notation is used by scientists, mathematicians, and engineers when they are working with very large or very small numbers. When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1.

 

Scientific Notation

A positive number is written in scientific notation if it is written as [latex]a\times10^{n}[/latex] where the coefficient a is [latex]1\leq{a}<10[/latex], and n is an integer.

Some calculators and other devices use [latex]E[/latex] instead of [latex]\times 10[/latex]. [latex]E[/latex] represents an exponent of [latex]10[/latex], then it is followed by a number which is the exponent used in scientific notation.

For example, [latex]1 \times 10^5 = 1E5[/latex].

Example 1

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.

1. Find the probability that the sum of the 80 values (or the total of the 80 values) is more than 7,500.

 

2. Find the sum that is 1.5 standard deviations above the mean of the sums.

Try it 1

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.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

To find percentiles for sums on the calculator, follow these steps.

2nd DIStR
3:invNormk = invNorm (area to the left of k, (n)(mean), (standard deviation))

where:

  • k is the kth percentile,
  • mean is the mean of the original distribution,
  • standard deviation is the standard deviation of the original distribution, and
  • sample size = n

Example 2

In a recent study reported October 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.

1. What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution?

 

2. Find the probability that the sum of the ages is between 1,500 and 1,800 years.

 

3. Find the 80th percentile for the sum of the 50 ages.

Try it 2

In a recent study reported October 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.

1. What are the mean and standard deviation for the sum of the ages of tablet users? What is the distribution?

 

2. Find the probability that the sum of the ages is between 1,400 and 1,500 years.

 

3. Find the 90th percentile for the sum of the 39 ages.

Example 3

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. What are the mean and standard deviation for the sums?

 

2. Find the 95th percentile for the sum of the sample. Interpret this value in a complete sentence.

 

3. Find the probability that the sum of the sample is at least ten hours.

Try it 3

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.

1. What is the probability that the sum of the sample is between seven hours and ten hours? What does this mean in context of the problem?

 

2. Find the 84th and 16th percentiles for the sum of the sample. Interpret these values in context.