The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 241 hours and a standard deviation of 12 hours. Find the first quartile, Q1.
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To find the first quartile, Q1, we need to find the value that separates the bottom 25% of the distribution from the rest.
Step 1: Convert the problem into Z-scores.
To do this, we can use the formula:
Z = (X - μ) / σ
Where:
X = the value we want to convert to a Z-score
μ = the mean of the distribution
σ = the standard deviation of the distribution
In this case, we want to find the Z-score that corresponds to the first quartile, Q1.
Step 2: Find the Z-score that corresponds to the first quartile.
The first quartile represents the 25th percentile. We can find the Z-score using a standard normal distribution table or calculator.
Using the Z-score formula:
Z = (Q1 - μ) / σ
To find Q1, we rearrange the equation:
Q1 = μ + Z * σ
Step 3: Substitute the given values into the equation to find Q1.
μ = 241 (mean of the distribution)
σ = 12 (standard deviation of the distribution)
Z = -0.674 (Z-score corresponding to the first quartile, obtained from the standard normal distribution table or calculator)
Q1 = 241 + (-0.674) * 12
Q1 = 241 - 8.088
Q1 ≈ 232.912
Therefore, the first quartile, Q1, is approximately 232.912 hours.
To find the first quartile, Q1, which represents the 25th percentile of the distribution, we need to use the Standard Normal Distribution table or a statistical calculator.
Here are the steps to find Q1:
Step 1: Convert the given data to the standard normal distribution by using the z-score formula:
z = (x - μ) / σ
Where,
x = raw score (value you want to convert)
μ = mean of the distribution
σ = standard deviation of the distribution
In this case:
x = Q1 (value we want to find)
μ = 241 (mean)
σ = 12 (standard deviation)
Step 2: Calculate the z-score for Q1 using the formula:
z = (Q1 - μ) / σ
Step 3: Use the z-score to find the cumulative probability associated with it. This gives us the percentage of data below Q1.
Step 4: Look up the cumulative probability from the standard normal distribution table or use a statistical calculator to find the z-score that corresponds to it.
Step 5: Once you have the z-score, use the z-score formula to find Q1:
Q1 = z * σ + μ
By following these steps, we can find the value for Q1.