6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:49 minutesProblem 7.CR.7b
Textbook Question
Normal Distribution Using a larger data set than the one given for the preceding exercises, assume that cell phone radiation amounts are normally distributed with a mean of 1.17 W/kg and a standard deviation of 0.29 W/kg.
b. Find the value of Q3, the cell phone radiation amount that is the third quartile.
Verified step by step guidance1
Step 1: Understand the problem. The third quartile (Q3) in a normal distribution corresponds to the 75th percentile. This means we need to find the value of the random variable (cell phone radiation amount) such that 75% of the data lies below it.
Step 2: Recall the formula for standardizing a value in a normal distribution: z = (x - μ) / σ, where z is the z-score, x is the value of interest, μ is the mean, and σ is the standard deviation. Here, μ = 1.17 W/kg and σ = 0.29 W/kg.
Step 3: Use a z-score table or a statistical tool to find the z-score corresponding to the 75th percentile. From standard normal distribution tables, the z-score for the 75th percentile is approximately z = 0.674.
Step 4: Rearrange the z-score formula to solve for x (the value of Q3): x = z * σ + μ. Substitute the known values: z = 0.674, μ = 1.17, and σ = 0.29.
Step 5: Perform the calculation to find Q3. This will give you the cell phone radiation amount corresponding to the third quartile.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
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. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the cell phone radiation amounts follow a normal distribution with a specified mean and standard deviation.
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Quartiles
Quartiles are values that divide a data set into four equal parts, each containing 25% of the data. The third quartile (Q3) is the value below which 75% of the data fall. It is a measure of statistical dispersion and helps in understanding the spread and central tendency of the data, particularly in a normal distribution.
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Calculating Q3 in Normal Distribution
To find Q3 in a normal distribution, one can use the z-score corresponding to the 75th percentile, which is approximately 0.674. The formula to calculate Q3 is Q3 = mean + (z-score * standard deviation). By substituting the given mean and standard deviation into this formula, one can determine the value of Q3 for the cell phone radiation amounts.
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