Respuesta :
Answer:
a. 0.691
b. 0.382
c. 0.933
d. $88.490
e. $58.168
f. 5th percentile: $42.103
95th percentile: $107.897
Step-by-step explanation:
We have, for the purchase amounts by customers, a normal distribution with mean $75 and standard deviation of $20.
a. This can be calculated using the z-score:
[tex]z=\dfrac{X-\mu}{\sigma}=\dfrac{85-75}{20}=\dfrac{10}{20}=0.5\\\\\\P(X<85)=P(z<0.5)=0.691[/tex]
The probability that a randomly selected customer spends less than $85 at this store is 0.691.
b. We have to calculate the z-scores for both values:
[tex]z_1=\dfrac{X_1-\mu}{\sigma}=\dfrac{65-75}{20}=\dfrac{-10}{20}=-0.5\\\\\\z_2=\dfrac{X_2-\mu}{\sigma}=\dfrac{85-75}{20}=\dfrac{10}{20}=0.5\\\\\\\\P(65<X<85)=P(-0.5<z<0.5)=P(z<0.5)-P(z<-0.5)\\\\P(65<x<85)=0.691-0.309=0.382[/tex]
The probability that a randomly selected customer spends between $65 and $85 at this store is 0.382.
c. We recalculate the z-score for X=45.
[tex]z=\dfrac{X-\mu}{\sigma}=\dfrac{45-75}{20}=\dfrac{-30}{20}=-1.5\\\\\\P(X>45)=P(z>-1.5)=0.933[/tex]
The probability that a randomly selected customer spends more than $45 at this store is 0.933.
d. In this case, first we have to calculate the z-score that satisfies P(z<z*)=0.75, and then calculate the X* that corresponds to that z-score z*.
Looking in a standard normal distribution table, we have that:
[tex]P(z<0.67449)=0.75[/tex]
Then, we can calculate X as:
[tex]X^*=\mu+z^*\cdot\sigma=75+0.67449\cdot 20=75+13.4898=88.490[/tex]
75% of the customers will not spend more than $88.49.
e. In this case, first we have to calculate the z-score that satisfies P(z>z*)=0.8, and then calculate the X* that corresponds to that z-score z*.
Looking in a standard normal distribution table, we have that:
[tex]P(z>-0.84162)=0.80[/tex]
Then, we can calculate X as:
[tex]X^*=\mu+z^*\cdot\sigma=75+(-0.84162)\cdot 20=75-16.8324=58.168[/tex]
80% of the customers will spend more than $58.17.
f. We have to calculate the two points that are equidistant from the mean such that 90% of all customer purchases are between these values.
In terms of the z-score, we can express this as:
[tex]P(|z|<z*)=0.9[/tex]
The value for z* is ±1.64485.
We can now calculate the values for X as:
[tex]X_1=\mu+z_1\cdot\sigma=75+(-1.64485)\cdot 20=75-32.897=42.103\\\\\\X_2=\mu+z_2\cdot\sigma=75+1.64485\cdot 20=75+32.897=107.897[/tex]
5th percentile: $42.103
95th percentile: $107.897
