X is distributed with mean 6 and standard deviation 2 and Y=2X+5. Find the mean and standard deviation of the Y distribution.
1 answer:
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Answer:
a) The range is (1199, 1267)
b) The range is (1165, 1301)
c) The range is (1131, 1335)
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:
Where and
The empirical rule, also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ). Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).
Part a
For this case we can use the statement from the empirical rule "68% of the data falls within the first standard deviation (µ ± σ)", and we can find the limits like this:
The range is (1199, 1267)
Part b
For this case we can use the statement from the empirical rule "95% of the data within the first two standard deviations (µ ± 2σ)", and we can find the limits like this:
The range is (1165, 1301)
Part c
For this case we can use the statement from the empirical rule "99.7% of the data within the first three standard deviations (µ ± 3σ)" and that represent almost all the data, and we can find the limits like this:
The range is (1131, 1335)