Respuesta :
Answer:
a) 0.9744; 0.4744; b) 36
Step-by-step explanation:
The formula for a z score of a mean of a sample is
[tex]z=\frac{\bar{X}-\mu}{\sigma \div \sqrt{n}}[/tex]
For part a,
Our mean, μ, is 2.66 and our standard deviation, σ, is 0.87. Our sample size, n, is 25. We want P(X ≤ 3.00):
z = (3.00-2.66)/(0.87÷√25)
z = 0.34/(0.87÷5)
z = 0.34/0.174 = 1.95
Using a z chart, we see that the area to the left of this value is 0.9744.
For the second question,
We want P(2.66 ≤ X ≤ 3.00):
z = (2.66-2.66)/(0.87÷√25) = 0/0.174 = 0
z = (3.00-2.66)/(0.87÷√25) = 0.34/0.174 = 1.95
The area under the curve to the left of z = 0 is 0.5000; the area under the curve to the left of z = 1.95 is 0.9744. This means the area between them is
0.9744-0.5000 = 0.4744.
For part b,
We want our probability to be at least 0.99. Since we have the area under the curve to the left in a z table, we will subtract from 1:
1-0.99 = 0.01
In the z table, we see the cell with the value closest to 0.01 is 0.0099; this corresponds with a z score of -2.33:
[tex]-2.33=\frac{3.00-2.66}{0.87\div \sqrt{n}}\\\\-2.33 = \frac{0.34}{0.87\div \sqrt{n}}\\\\-2.33 = 0.34 \div \frac{0.87}{\sqrt{n}}\\\\-2.33 = 0.34 \times \frac{\sqrt{n}}{0.87}\\\\-2.33 = \frac{0.34\sqrt{n}}{0.87}[/tex]
Multiply both sides by 0.87:
0.87(-2.33) = ((0.34√n)/0.87)
-2.0271 = 0.34√n
Divide both sides by 0.34:
-2.0271/0.34 = 0.34√n/0.34
-5.9621 = √n
Square both sides:
(-5.9621)² = (√n)²
35.55 = n
This rounds to 36.
The probability that the sample average sediment density is at most 3.00 is 0.4747.
How to calculate probability?
From the information given, the mean is 2.66 and the standard deviations is given as 0.87. Therefore, the corresponding z value will be calculates thus:
= P( Z < [(3 - 2.66) / (0.87 / ✓25)]
= P(Z < 1.954)
= 0.9747
Therefore, the probability that the sample average sediment density is at most 3.00 will be:
= 0.9747 - 0.5
= 0.4747
Also, the sample size that would be required to ensure that the probability in part is at least 0.99 will be:
= 0.34 / (0.87/✓n) > 2.33
✓n > 35.5
n = 36
In conclusion, the sample is 36.
Learn more about probability on:
https://brainly.com/question/25870256
