Respuesta :
Answer:
a) P(x > 43) = 0.9599
b) P(x < 42) = 0.0228
c) P(x > 57.5) = 0.03
d) P(x = 42) = 0.
e) P(x<40 or x>55) = 0.1118
f) 43.42
g) Between 46.64 and 53.36.
h) Above 45.852.
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this question, we have that:
[tex]\mu = 50, \sigma = 4[/tex]
a) x>43
This is 1 subtracted by the pvalue of Z when X = 43. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{43 - 50}{4}[/tex]
[tex]Z = -1.75[/tex]
[tex]Z = -1.75[/tex] has a pvalue of 0.0401
1 - 0.0401 = 0.9599
P(x > 43) = 0.9599
b) x<42
This is the pvalue of Z when X = 42.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{42 - 50}{4}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] has a pvalue of 0.0228
P(x < 42) = 0.0228
c) x>57.5
This is 1 subtracted by the pvalue of Z when X = 57.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{57.5 - 50}{4}[/tex]
[tex]Z = 1.88[/tex]
[tex]Z = 1.88[/tex] has a pvalue of 0.97
1 - 0.97 = 0.03
P(x > 57.5) = 0.03
d) P(x = 42)
In the normal distribution, the probability of an exact value is 0. So
P(x = 42) = 0.
e) x<40 or x>55
x < 40 is the pvalue of Z when X = 40. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{40 - 50}{4}[/tex]
[tex]Z = -2.5[/tex]
[tex]Z = -2.5[/tex] has a pvalue of 0.0062
x > 55 is 1 subtracted by the pvalue of Z when X = 55. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{55 - 50}{4}[/tex]
[tex]Z = 1.25[/tex]
[tex]Z = 1.25[/tex] has a pvalue of 0.8944
1 - 0.8944 = 0.1056
0.0062 + 0.1056 = 0.1118
P(x<40 or x>55) = 0.1118
f) 5% of the values are less than what X value?
X is the 5th percentile, which is X when Z has a pvalue of 0.05, so X when Z = -1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 50}{4}[/tex]
[tex]X - 50 = -1.645*4[/tex]
[tex]X = 43.42[/tex]
43.42 is the answer.
g) 60% of the values are between what two X values (symmetrically distributed around the mean)?
Between the 50 - (60/2) = 20th percentile and the 50 + (60/2) = 80th percentile.
20th percentile:
X when Z has a pvalue of 0.2. So X when Z = -0.84.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.84 = \frac{X - 50}{4}[/tex]
[tex]X - 50 = -0.84*4[/tex]
[tex]X = 46.64[/tex]
80th percentile:
X when Z has a pvalue of 0.8. So X when Z = 0.84.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.84 = \frac{X - 50}{4}[/tex]
[tex]X - 50 = 0.84*4[/tex]
[tex]X = 53.36[/tex]
Between 46.64 and 53.36.
h) 85% of the values will be above what X value?
Above the 100 - 85 = 15th percentile, which is X when Z has a pvalue of 0.15. So X when Z = -1.037.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.037 = \frac{X - 50}{4}[/tex]
[tex]X - 50 = -1.037*4[/tex]
[tex]X = 45.852[/tex]
Above 45.852.
