Respuesta :
Answer:
25.10% probability that the spending is between 46 and 49.56 dollars
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 = 47.67, \sigma = 5.5[/tex]
What is the probability that the spending is between 46 and 49.56 dollars?
This is the pvalue of Z when X = 49.56 subtracted by the pvalue of Z when X = 46. So
X = 49.56
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{49.56 - 47.67}{5.5}[/tex]
[tex]Z = 0.34[/tex]
[tex]Z = 0.34[/tex] has a pvalue of 0.6331
X = 46
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{46 - 47.67}{5.5}[/tex]
[tex]Z = -0.3[/tex]
[tex]Z = -0.3[/tex] has a pvalue of 0.3821
0.6331 - 0.3821 = 0.2510
25.10% probability that the spending is between 46 and 49.56 dollars
