A heavy rope, 60 ft long, weighs 0.3 lb/ft and hangs over the edge of a building 130 ft high. (Let x be the distance in feet bel
1 answer:
Answer:
half the rop to the top of the building = W = 423.75
top of the building = W = 423.75
Step-by-step explanation:
a) How much work W is done in pulling the rope to the top of the building?
When a length of rope xi has been pulled to the top of the building, then the force required to move the remaining rope is 1/2(g(60− xi)), the weight of the rope.
The work to pull an additional small length of rope, ∆x, to the top of the building is 1 /2 ((60 − xi)∆x).
Note that the relationship between xi and ∆x is that for n constant lengths of rope equal to ∆x, then xi = i∆x. The Riemann sum for the work is:
W= ∑ n,i = 1 (1/2(60 - xi). ∆x) = ∑ n,i = 1 (1/2(60 - i∆x). ∆x)
Note that xi = i∆x will not limit to zero as n → ∞. The corresponding integral is:
W =
W =
W = 423.75
d) How much work W is done in pulling half the rope to the top of the building?
This requires almost the same Riemann sum except the upper limit on the sum changes:
W= ∑ n/2,i = 1 (1/2(60 - xi). ∆x) = ∑ n,i = 1 (1/2(60 - i∆x). ∆x)
The corresponding integral is:
W =
W =
W = 423.75
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Answer:
a) Null Hypothesis:
Alternative hypothesis:
b) The 95% confidence interval would be given by (910.05;959.95)
c) Since we confidence interval not ocntains the value of 900 we fail to reject the null hypothesis that the true mean is 900.
d)
Since is a bilateral test the p value is given by:
Step-by-step explanation:
a. State the hypotheses.
On this case we want to check the following system of hypothesis:
Null Hypothesis:
Alternative hypothesis:
b. What is the 95% confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean x¯¯¯= 935?
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".
represent the sample mean for the sample
population mean (variable of interest)
represent the population standard deviation
n=200 represent the sample size
The confidence interval for the mean is given by the following formula:
(1)
In order to calculate the mean and the sample deviation we can use the following formulas:
(2)
(3)
The mean calculated for this case is
The sample deviation calculated
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 (910.05;959.95)
c. Use the confidence interval to conduct a hypothesis test. Using α= .05, what is your conclusion?
Since we confidence interval not ocntains the value of 900 we fail to reject the null hypothesis that the true mean is 900.
d. What is the p-value?
The statistic is given by:
If we replace we got:
Since is a bilateral test the p value is given by:
So then since the p value is less than the significance we can reject the null hypothesis at 5% of significance.