A certain book is for sale in both Belize and Brazil. In Belize, it sells for 80.39 Belize dollars. In Brazil, it sells for 63.6
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Answer:
The blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L is 3.64.
Step-by-step explanation:
To solve this question, we have to understand the normal probability distribution and the central limit theorem.
Normal probability distribution:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random variable X, with mean and standard deviation
, the sample means of size n can be approximated to a normal distribution with mean
and standard deviation, which is also called standard error
In this problem, we have that:
What is the blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L?
This is the value of X when Z has a pvalue of 0.05. So it is X when Z = -1.645.
By the Central Limit Theorem
The blood potassium level L such that the probability is only 0.05 that the average of four measurements is less than L is 3.64.
Answer:
And if we solve for a we got
So the value of height that separates the bottom 92% of data from the top 8% is 649.765.
Step-by-step explanation:
Previous concepts
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".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:
Where and
For this part we want to find a value a, such that we satisfy this condition:
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.92 of the area on the left and 0.08 of the area on the right it's z=1.405. On this case P(Z<1.405)=0.92 and P(z>0.92)=0.08
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got
So the value of height that separates the bottom 92% of data from the top 8% is 649.765.
Chebyshev's theorem in statistics states that for many probability distributions, no more than 1/k² of measured values will be k standard deviations away from the mean.
Because the area under the probability distribution curve is equal to 1, Chebyshev's theorem means that the shaded area shown in the figure is equal to 1 - 1/k².
When k = 1.75, the shaded area is
1 - 1/1.75² = 0.7635 = 67.35%
Therefore the percent of the area within +/- 1.75 standard deviations from the mean is
67.35/2 = 33.7%, which is at least 33% of the observations.
Answer:
According to the Chebyshev theorem, at least 33% of the observations lie within +/- standard deviations from the mean.
Because the area under the probability distribution curve is equal to 1, Chebyshev's theorem means that the shaded area shown in the figure is equal to 1 - 1/k².
When k = 1.75, the shaded area is
1 - 1/1.75² = 0.7635 = 67.35%
Therefore the percent of the area within +/- 1.75 standard deviations from the mean is
67.35/2 = 33.7%, which is at least 33% of the observations.
Answer:
According to the Chebyshev theorem, at least 33% of the observations lie within +/- standard deviations from the mean.