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<em>What you need to find is the extra amount of plywood. To do that, you must find the area of the square and the area of the circle and subtract them to find the answer.</em>
Formulas:
area of a square: A = bh
area of a circle: A =
square measurements:
b = 28
h = 28
circle measurements:
(≈3.14)
r = 14
A = (28)(28)
A = 784
area of square = 784 in^2
A = 3.14 (
)
A = 3.14 (196)
A = 615.44 in^2
784 - 615.44 = 168.56
The amount of plywood left over is 168.56 in^2.
Formulas:
area of a square: A = bh
area of a circle: A =
square measurements:
b = 28
h = 28
circle measurements:
r = 14
A = (28)(28)
A = 784
area of square = 784 in^2
A = 3.14 (
A = 3.14 (196)
A = 615.44 in^2
784 - 615.44 = 168.56
The amount of plywood left over is 168.56 in^2.
Answer:
a) The standard deviation of this sampling distribution is 2.07.
b) The missing number is 4.14.
c) The 95% confidence interval for the population mean score μ based on this one sample is between 267.86 and 276.14.
Step-by-step explanation:
To solve this question, we need to understand the Empirical Rule and the Central Limit Theorem.
Empirical Rule:
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
Central Limit Theorem:
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question:
(a) If we take many samples, the sample mean x⎯⎯⎯ varies from sample to sample according to a Normal distribution with mean equal to the unknown mean score μ in the population. What is the standard deviation of this sampling distribution?
Using the Central Limit Theorem:
The standard deviation of this sampling distribution is 2.07.
(b) According to the 95 part of the 68-95-99.7 rule, 95% of all values of x⎯⎯⎯ fall within _______ on either side of the unknown mean μ. What is the missing number?
Within 2 standard deviations of the mean.
So, 2*2.07 = 4.14
The missing number is 4.14.
(c) What is the 95% confidence interval for the population mean score μ based on this one sample?
Within 4.14 of the mean
272 - 4.14 = 267.86
272 + 4.14 = 276.14
The 95% confidence interval for the population mean score μ based on this one sample is between 267.86 and 276.14.