Respuesta :
Answer:
0.7325 to 5.6675 ug/dl
Step-by-step explanation:
The middle 90% will be 45% above the mean and 45% below the mean. This means
0.5-0.45 = 0.05 and
0.5+0.45 = 0.95
We use a z table. Look in the cells; find the values as close to 0.05 and 0.95 as we can get.
For 0.05, we have 0.0505 and 0.0495; since these are equidistant from 0.05, we use the value between them. 0.0505 is z=-1.64 and 0.0495 is z=1.65; this gives us z=-1.645.
For 0.95, we have 0.9495 and 0.9505; since these are equidistant from 0.95, we use the value between them. 0.9495 is z = 1.64 and 0.9505 is z=1.65; this gives us z = 1.645.
Now we use our z score formula,
[tex]z=\frac{X-\mu}{\sigma}[/tex]
Our two z scores are 1.645 and -1.645; our mean, μ, is 3.2; and our standard deviation, σ, is 1.5:
[tex]1.645=\frac{X-3.2}{1.5}[/tex]
Multiply both sides by 1.5:
[tex]1.5(1.645)=\frac{X-3.2}{1.5}\times 1.5\\\\2.4675 = X-3.2[/tex]
Add 3.2 to each side:
2.4675+3.2 = X-3.2+3.2
5.6675 = X
[tex]-1.645=\frac{X-3.2}{1.5}[/tex]
Multiply both sides by 1.5:
[tex]1.5(-1.645)=\frac{X-3.2}{1.5}\times 1.5\\\\-2.4675=X-3.2[/tex]
Add 3.2 to each side:
-2.4675+3.2 = X-3.2+3.2
0.7325 = X
Our range is from 0.7325 to 5.6675.
Using the normal distribution, it is found that the middle 90% is between 0.26 and 6.14 ug/dl.
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- In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It 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, which is the percentile of X.
In this problem:
- The mean is of 3.5 ug/dl, thus [tex]\mu = 3.5[/tex].
- The standard deviation is of 1.5 ug/dl, thus [tex]\sigma = 1.5[/tex].
- The normal distribution is symmetric, which means that the middle 90% is between the 5th percentile and the 95th percentile.
- The 5th percentile is X when Z has a p-value of 0.05, so X when Z = -1.96.
- The 95th percentile is X when Z has a p-value of 0.95, so X when Z = 1.96.
5th percentile:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.96 = \frac{X - 3.2}{1.5}[/tex]
[tex]X - 3.2 = -1.96(1.5)[/tex]
[tex]X = 0.26[/tex]
95th percentile:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.96 = \frac{X - 3.2}{1.5}[/tex]
[tex]X - 3.2 = 1.96(1.5)[/tex]
[tex]X = 6.14[/tex]
The range of the middle 90% is between 0.26 and 6.14 ug/dl.
A similar problem is given at https://brainly.com/question/23285700