Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sample means of size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation, which is also called standard error [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 3.8, \sigma = 0.2, n = 4, s = \frac{0.2}{\sqrt{4}} = 0.1[/tex]
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.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-1.645 = \frac{X - 3.8}{0.1}[/tex]
[tex]X - 3.8 = -1.645*0.1[/tex]
[tex]X = 3.64[/tex]
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:
Blood potassium level L = 3.64
Step-by-step explanation:
We are given that Judy’s measured potassium level varies according to the Normal distribution with μ = 3.8 and σ = 0.2 mm .
Let X bar = average of four measurements
So, X bar ~ N([tex]\mu = 3.8,\sigma = 0.2[/tex])
The z score probability distribution of sample mean is ;
Z = [tex]\frac{Xbar -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, n = sample size = 4
[tex]\mu[/tex] = population mean = 3.8
[tex]\sigma[/tex] = population standard deviation = 0.2
The condition given to us is that probability that the average of four measurements is less than L is only 0.05 , i.e.;
P(X bar < L) = 0.05
P( [tex]\frac{Xbar -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{L -3.8}{\frac{0.2}{\sqrt{4} } }[/tex] ) = 0.05
P(Z < [tex]\frac{L -3.8}{0.1 }[/tex] ) = 0.05
In z table, we find that the critical value below which the z probability is 0.05 is given as -1.6449 .
So, it means; [tex]\frac{L -3.8}{0.1 }[/tex] = -1.6449
L = (-1.6449 * 0.1) + 3.8
L = -0.16449 + 3.8 = 3.64
Therefore, 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 .
