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The Central Limit Theorem for Sums: ∑X ~ N [( n )( μ x ),( )( σ x )]
Mean for Sums ( ∑X ): ( n )( μ x )
The Central Limit Theorem for Sums z -score and standard deviation for sums:
Standard deviation for Sums ( ∑X ): ( σ x )
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 probability that the sum of the 100 values falls between the numbers you found in [link] and [link] .
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 exercise: 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 P ( Σx = 290)?
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 = 1,186?
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 P ( Σx >9,000)?
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