Respuesta :
Answer:
69.27% probability that she is between 60 and 66 inches
Step-by-step explanation:
Problems of normally distributed samples are solved using 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 problem, we have that:
[tex]\mu = 64.3, \sigma = 2.6[/tex]
If a woman is selected at random from the study, what is the chance that she is between 60 and 66 inches
This is the pvalue of Z when X = 66 subtracted by the pvalue of Z when X = 60. So
X = 66
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{66 - 64.3}{2.6}[/tex]
[tex]Z = 0.65[/tex]
[tex]Z = 0.65[/tex] has a pvalue of 0.7422
X = 60
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{60 - 64.3}{2.6}[/tex]
[tex]Z = -1.65[/tex]
[tex]Z = -1.65[/tex] has a pvalue of 0.0495.
0.7422 - 0.0495 = 0.6927
69.27% probability that she is between 60 and 66 inches
Answer: P(60 ≤ x ≤ 66) = 0.69
Step-by-step explanation:
Since the heights were approximately normally distributed, we would apply the formula for normal distribution which is expressed as
z = (x - µ)/σ
Where
x = heights of the women
µ = mean height
σ = standard deviation
From the information given,
µ = 64.3 inches
σ = 2.6 inches
The the chance that she is between 60 and 66 inches is expressed as
P(60 ≤ x ≤ 66)
For x = 60,
z = (60 - 64.3)/2.6 = - 1.65
Looking at the normal distribution table, the probability corresponding to the z score is 0.05
For x = 66,
z = (66 - 64.3)/2.6 = 0.65
Looking at the normal distribution table, the probability corresponding to the z score is 0.74
Therefore,
P(60 ≤ x ≤ 66) = 0.74 - 0.05 = 0.69
