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sweetcandy16gaming
18.02.2020 •
Mathematics
The amount of time that people spend at Grover Hot Springs Spa is normally distributed with a mean of 63 minutes and a standard deviation of 18 minutes. Answer the following, rounding probabilities to 4 decimal places. Let X= the amount of time that a person spends at Grover Hot Springs Spa a. The distribution is X ~ ( , ) b. What's the probability that a randomly chosen person spends more than 90 minutes at the Spa ?c. What's the probability that a randomly person spends less than 45 minutes at the Spa?d. What's the probability that a randomly person spends between 60 and 90 minutes at the Spa?e. Find the IQR. minutes Round answers to 2 decimals.
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Ответ:
a)
Where
and ![\sigma=18](/tpl/images/0514/5199/76e06.png)
b)![P(X90)=P(\frac{X-\mu}{\sigma}\frac{90-\mu}{\sigma})=P(Z\frac{90-63}{18})=P(Z1.5)](/tpl/images/0514/5199/59c21.png)
And we can find this probability using the complement rule and with the normal standard table or excel:
c)![P(X](/tpl/images/0514/5199/18569.png)
d)![P(60](/tpl/images/0514/5199/f714f.png)
e) IQR = 75.13-50.87=24.26
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the amount of time that people spend at Grover Hot Springs of a population, and for this case we know the distribution for X is given by:
Where
and ![\sigma=18](/tpl/images/0514/5199/76e06.png)
Part b
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
And we can find this probability using the complement rule and with the normal standard table or excel:
Part c
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
And we can find this probability using the normal standard table or excel:
Part d
If we apply this formula to our probability we got this:
And we can find this probability with this difference:
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
Part e
Q1
For this part we want to find a value a, such that we satisfy this condition:
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.25 of the area on the left and 0.75 of the area on the right it's z=-0.674. On this case P(Z<-0.674)=0.25 and P(z>-0.674)=0.75
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 25% of data from the top 75% is 50.87.
Q3
Since the distribution is symmetrical we repaeat the procedure for Q1 but now with z= 0.674
And if we solve for a we got
And then the IQR = 75.13-50.87=24.26
Ответ:
,loiuytgfvbn878987890;lkjhygtfds
Step-by-step explanation: