In a recent study on world happiness, participants were asked to evaluate their current lives on a scale of 0 to 10, where 0 represents the worst possible life and 10 represents the best possible life. The mean response was 5.9 with a standard deviation of 2.6.
a. what response represents the 88th percentile?
b. what response represents the 60th percentile?
c. what response represents the first quartile?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability and its Z score.
Z = (score-mean)/SD
Insert the data into the equation to calculate the score (response).
To find the responses for the given percentiles, we need to use the concept of z-scores. Z-scores allow us to determine the position of a data point in relation to the mean and standard deviation.
a. To find the response for the 88th percentile, we need to find the corresponding z-score first. The formula for the z-score is z = (x - μ) / σ, where x represents the data point, μ represents the mean, and σ represents the standard deviation.
To find the z-score for the 88th percentile, we need to find the value of x. Rearranging the formula, x = z * σ + μ. Plug in the values:
z = InvNorm(0.88) (using a standard normal distribution table or a calculator)
σ = 2.6 (standard deviation)
μ = 5.9 (mean)
By calculating the values, we find:
z = InvNorm(0.88) = 1.174
x = 1.174 * 2.6 + 5.9 = 8.53
Therefore, the response that represents the 88th percentile is approximately 8.53.
b. To find the response for the 60th percentile, we will follow the same process. Again, we need to find the corresponding z-score and use the formula x = z * σ + μ.
z = InvNorm(0.60)
σ = 2.6
μ = 5.9
By calculating the values, we find:
z = InvNorm(0.60) = 0.253
x = 0.253 * 2.6 + 5.9 = 6.51
Therefore, the response that represents the 60th percentile is approximately 6.51.
c. To find the response that represents the first quartile, we need to find the z-score that corresponds to the 25th percentile. We can then use the same formula x = z * σ + μ.
z = InvNorm(0.25)
σ = 2.6
μ = 5.9
By calculating the values, we find:
z = InvNorm(0.25) = -0.674
x = -0.674 * 2.6 + 5.9 = 4.806
Therefore, the response that represents the first quartile is approximately 4.806.
To find the response that represents a particular percentile, we need to use the concept of Z-scores. A Z-score tells us how many standard deviations a particular value is away from the mean.
a. To find the response that represents the 88th percentile, we can start by finding the Z-score corresponding to this percentile.
Formula for Z-score: (X - Mean) / Standard Deviation
Z = (X - 5.9) / 2.6
To find the X value (response) corresponding to the 88th percentile, we need to find the Z-score that corresponds to the 88th percentile. We can do this by using a Z-table or a statistical calculator. From the Z-table, we can see that the Z-score for the 88th percentile is approximately 1.18.
Now, we can rearrange the formula for Z-score to solve for X:
X = (Z * Standard Deviation) + Mean
X = (1.18 * 2.6) + 5.9
Calculating the value, we get:
X ≈ 8.068
So, the response that represents the 88th percentile is approximately 8.068.
b. To find the response that represents the 60th percentile, we follow the same process as in part a. First, we find the Z-score for the 60th percentile. From the Z-table, we find that the Z-score for the 60th percentile is approximately 0.25.
Using the formula for X, we have:
X = (0.25 * 2.6) + 5.9
Calculating the value, we get:
X ≈ 6.275
So, the response that represents the 60th percentile is approximately 6.275.
c. The first quartile represents the 25th percentile. To find the response that represents the first quartile, we find the Z-score for the 25th percentile. From the Z-table, we find that the Z-score for the 25th percentile is approximately -0.68.
Using the formula for X:
X = (-0.68 * 2.6) + 5.9
Calculating the value, we get:
X ≈ 4.132
So, the response that represents the first quartile is approximately 4.132.