Use the descriptive statistics (below) for the variable “height” from the 2014 G

Question

Use the descriptive statistics (below) for the variable “height” from the 2014 General Social Survey and a z-score table to answer questions #1-7. Show your work. Statistics R IS HOW TALL N Valid 1230 Missing 2612 Mean 67 Median 67 Std. Deviation 4 Minimum 57 Maximum 79 Find the z-score for someone 70 inches tall. What does it mean? Assuming that the distribution is normal, what proportion of the sample is 70 inches tall or shorter? Assuming that the distribution is normal, what proportion of the sample is 70 inches tall or taller? Find the z-score for someone 56 inches tall. What does it mean? Assuming that the distribution is normal, what proportion of the sample is 56 inches tall or shorter? Assuming that the distribution is normal, what proportion of the sample is 56 inches tall or taller? Assuming that the distribution is normal, what proportion of the sample is between 56 and 70 inches tall?

Explanation / Answer

(1) The z-score formula is given by : (x - mu)/sigma, where "x" is the original score, "mu" is the mean of the distribution and "sigma" is the standard deviation of the distribution.
Here, x = 70, mu = 67, sigma = 4.
Z-score = (70-67)/4 = 3/4 = 0.75
The z score tells us how many standard deviations from the mean our score is. Here, our original score, that is, 70 inches tall, is 0.75 standard deviations above the mean.

(2) Now to find proportion of the sample that is 70 inches tall or shorter = P(X <= 70).
Here, X is a normal distribution with mean = 67 and standard deviation = 4
Then, P(X <= 70) = 0.77 (approx.)

(3) Now to find proportion of the sample that is 70 inches tall or taller = P(X >= 70).
Then P(X >= 70) = 1 - P(X <= 70) = 0.23 (approx.)

(4) The z-score for someone who is 56 inches tall = (56-67)/4 = -2.75.
Here, the original score, that is, 56 inches tall, is 2.75 standard deviations below the mean.

(5) Proportion of sample that is 56 inches tall or shorter = P(X <= 56).
Here, X is normally distributed with mean 67 and sd 4.
Then, P(X <= 56) = 0.003 (approx.)

(6) Proportion of sample that is 56 inches tall or taller = P(X >= 56).
Then, P(X >= 56) = 0.997 (approx.)

(7) Proportion of sample that is between 56 and 70 inches tall = P(56 <= X <= 70) = P(X <= 70) - P(X <= 56) = 0.77 - 0.003 = 0.767 (approx.)

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