Hexagon ABCDEF is a regular hexagon. Show that segments AE and ED are perpendicular.
1 answer:
For a regular hexagon, the interior angles are 120°, since they add up to 720 (180(6 - 2))
We also know, for a regular hexagon, all sides are equal.
Refer to my diagram.
Triangle AFE is an isosceles triangle, because two sides of the triangle are equal in a regular hexagon.
Thus, 120 + 2x = 180 (angle sum of triangle)
2x = 60 and x = 30
Now, ∠FEA = 30°, but we know that ∠FED is 120° (all interior angles of a regular hexagon is 120°)
We know that ∠FED = ∠FEA + ∠AED
120° = 30° + ∠AED
∴ ∠AED = 90° (120 - 30)
If ∠AED is 90°, then, by definition, AE ⊥ ED
We also know, for a regular hexagon, all sides are equal.
Refer to my diagram.
Triangle AFE is an isosceles triangle, because two sides of the triangle are equal in a regular hexagon.
Thus, 120 + 2x = 180 (angle sum of triangle)
2x = 60 and x = 30
Now, ∠FEA = 30°, but we know that ∠FED is 120° (all interior angles of a regular hexagon is 120°)
We know that ∠FED = ∠FEA + ∠AED
120° = 30° + ∠AED
∴ ∠AED = 90° (120 - 30)
If ∠AED is 90°, then, by definition, AE ⊥ ED
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Answer:
0.007 = 0.7% probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean and standard deviation
, the z-score of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes 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.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation
A statistician calculates that 8% of Americans own a Rolls Royce.
This means that
Sample of 595:
This means that
Mean and standard deviation:
What is the probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%?
Proportion above 8% + 3% = 11% or below 8% - 3% = 5%. Since the normal distribution is symmetric, these probabilities are equal, and so we find one of them and multiply by 2.
Probability the proportion is less than 5%:
P-value of Z when X = 0.05. So
By the Central Limit Theorem
has a p-value of 0.0035
2*0.0035 = 0.0070
0.007 = 0.7% probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%