10x18+a what is the pythagorean theorem
1 answer:
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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.
Step 1 :
simplify
Equation at the end of step 1 :
15 (((3 • (x2)) - 4x) - ——) - 3 x
Step 2 :
Equation at the end of step 2 :
15 ((3x2 - 4x) - ——) - 3 x
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using x as the denominator :
3x2 - 4x (3x2 - 4x) • x 3x2 - 4x = ———————— = —————————————— 1 x
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
3x2 - 4x = x • (3x - 4)
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
x • (3x-4) • x - (15) 3x3 - 4x2 - 15 ————————————————————— = —————————————— x x
Equation at the end of step 4 :
(3x3 - 4x2 - 15) ———————————————— - 3 x
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x as the denominator :
3 3 • x 3 = — = ————— 1 x