Claire is a manager at a toy packaging company. The company packs 80 boxes of toys every hour for the first 3 hours of the day.
They stop packaging for the next 2 hours in order to carry out a training session. Then, for the next 4 hours, they pack 20 boxes of toys every hour. Which graph models the piecewise function for the given situation?
2 answers:
Answer:
See the attached figure.
Step-by-step explanation:
Modeling each part as a function.
Let x is hours of the working day
The company packs 80 boxes of toys every hour for the first 3 hours of the day. ⇒ y₁ = 80 x , x ∈ [0,3]
At the end of the 3 hours The company packs = 80 * 3 = 240
They stop packaging for the next 2 hours in order to carry out a training session. ⇒ y₂ = 240 , x ∈ [3,5]
Then, for the next 4 hours, they pack 20 boxes of toys every hour
y₃ = 20x + c , x ∈ [5,9]
To find c, y₂ = y₃ at x =5
∴ 240 = 20 * 5 + c
∴ c = 240 - 20 * 5 =240 - 100 = 140
∴ y₃ = 20x + 140 , x ∈ [5,9]
So,
y₁ = 80 x , x ∈ [0,3]
y₂ = 240 , x ∈ [3,5]
y₃ = 20x + 140 , x ∈ [5,9]
The graph models the piecewise function for the given situation is as shown in the attached figure.
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Answer:
a)
b) The 95% confidence interval would be given by (138.846;149.754)
c) n=34
d) n=298
e) n=1190
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
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".
We can calculate the sample mean with this formula:
represent the sample mean
population mean (variable of interest)
represent the population standard deviation
n=10 represent the sample size
a) Find a point estimate of the population mean
We can calculate the sample mean with this formula:
b) Construct a 95% confidence interval for the true mean score for this population.
The confidence interval for the mean is given by the following formula:
(1)
Since the Confidence is 0.95 or 95%, the value of and
, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that
Now we have everything in order to replace into formula (1):
So on this case the 95% confidence interval would be given by (138.846;149.754)
Part c
The margin of error is given by this formula:
(a)
And on this case we have that ME =+20 and we are interested in order to find the value of n, if we solve n from equation (a) we got:
(b)
The critical value for 95% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.025;0;1)", and we got , replacing into formula (5) we got:
So the answer for this case would be n=34 rounded up to the nearest integer
Part d
So the answer for this case would be n=298 rounded up to the nearest integer
Part e
So the answer for this case would be n=1190 rounded up to the nearest integer