Question

The mean of a population is 77 and the standard deviation is 14. The shape of the population is unknown. Determine the probability of each of the following occurring from this population. Appendix A Statistical Tables a. A random sample of size 33 yielding a sample mean of 78 or more b. A random sample of size 130 yielding a sample mean of between 76 and 79 c. A random sample of size 219 yielding a sample mean of less than 77.5 (Round all the values of z to 2 decimal places and final answers to 4 decimal places.)

Answer 1

a)
Here, μ = 77, σ = 2.4371 and x = 78. We need to compute P(X >= 78). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (78 - 77)/2.4371 = 0.41

Therefore,
P(X >= 78) = P(z <= (78 - 77)/2.4371)
= P(z >= 0.41)
= 1 - 0.6591
= 0.3409

b)
Here, μ = 77, σ = 1.2279, x1 = 76 and x2 = 79. We need to compute P(76<= X <= 79). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (76 - 77)/1.2279 = -0.81
z2 = (79 - 77)/1.2279 = 1.63

Therefore, we get
P(76 <= X <= 79) = P((79 - 77)/1.2279) <= z <= (79 - 77)/1.2279)
= P(-0.81 <= z <= 1.63) = P(z <= 1.63) - P(z <= -0.81)
= 0.9484 - 0.209
= 0.7394

c)
Here, μ = 77, σ = 0.946 and x = 77.5. We need to compute P(X <= 77.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (77.5 - 77)/0.946 = 0.53

Therefore,
P(X <= 77.5) = P(z <= (77.5 - 77)/0.946)
= P(z <= 0.53)
= 0.7019