Patients who have artificial hip-replacement surgery experience pain the first day after surgery. Typically, the pain is measured on a subjective scale using values of 1 to 5. Let x represent the random variable, the pain score as determined by the patient. The probability distribution for x is believed to be as below. (Give your answers correct to two decimal places.)
x 1 2 3 4 5
P(x) 0.09 0.14 0.21 0.17 0.39
(a) Find the mean of x.
Incorrect: Your answer is incorrect. .
Redone, added x which is 15/5=3.0
(b) Find the standard deviation of x.
added all of p(x) and then sqrt the answer which came up with 0.2
Xp(x) +...... = 3.63
sqrt((x^2(px) -3.63^2)) = 1.35
To find the mean (a.k.a. expected value) of the random variable x, which represents the pain score, you need to multiply each pain score by its corresponding probability and then sum them up.
So, the calculations would be as follows:
Mean (μ) = (1 * 0.09) + (2 * 0.14) + (3 * 0.21) + (4 * 0.17) + (5 * 0.39)
= 0.09 + 0.28 + 0.63 + 0.68 + 1.95
= 3.63
Therefore, the mean of the random variable x is 3.63.
Now, to find the standard deviation (σ) of x, you need to calculate the variance first. The variance can be found by subtracting the mean from each possible outcome squared, then multiplying it by its corresponding probability and summing them up.
So, the calculations for the variance would be:
Variance (σ²) = [(1 - 3.63)² * 0.09] + [(2 - 3.63)² * 0.14] + [(3 - 3.63)² * 0.21] + [(4 - 3.63)² * 0.17] + [(5 - 3.63)² * 0.39]
= (2.63² * 0.09) + (1.63² * 0.14) + (0.63² * 0.21) + (0.37² * 0.17) + (1.37² * 0.39)
= 1.4583 + 0.3561 + 0.0838 + 0.0563 + 0.7479
= 2.7024
Now, you can find the standard deviation by taking the square root of the variance:
Standard Deviation (σ) = √2.7024
= 0.2 (rounded to two decimal places)
Therefore, the standard deviation of the random variable x is 0.2.