Respuesta :
Part a: The probability that the sample average size is at most 3.0 is 0.9957 and the probability that the sample average size is between 2.6 and 3.0 is 0.4957
Part b: The minimum sample size required so that the probability that the sample average sediment density is at most 3.00 is 20
The standard error decreases as the sample size increases. It shows that as the sample size grows, the variance of the sample statistics lowers and there is a greater likelihood of discovering sample statistics that are near to the mean.
Part a:
The mean, mu = 2.67
The population standard deviation, sigma = 0.75
The sample size, n = 25
z score:
[tex]z= \dfrac{(\bar{x}-\mu )}{ \sigma_{\bar{x}} }[/tex]
The standard error:
[tex]{ \sigma_{\bar{x}}= {\dfrac{\sigma{} }{\sqrt{n} }[/tex]
= 0.75/[tex]\sqrt{25\\}[/tex]
= 0.15
The probability that the sample average sediment density is at most 3.00:
P(xˉ≤3)=P(z≤3−2.60/0.152)
=P(z≤2.63)
=0.9957
The probability that the sample average sediment density is between 2.6 and 3.00:
P(2.6<xˉ<3.0)=P(2.6−2.6/0.152<z<3.0−2.6/0.152)
=P(0<z<2.63)
=P(z<2.63)−P(z<0)
=0.4957
Part b:
The sample size required so that the probability in part (a) is at least 0.99:
[tex]P(\bar x \leq 3) = 0.99 \\[1ex] P(Z \leq z) = 0.99 \\[1ex] z = 2.33[/tex]
Using the z-distribution table.
We know:
[tex]z = \dfrac{\bar x-\mu}{\sigma_{\bar x}}\\[2ex] \sigma_{\bar x} = \dfrac{\bar x-\mu}{z}[/tex]
When x = 3 and z = 2.33
σxˉ=xˉ−μ/z
=3−2.6/2.33
=0.1717
The standard error:
[tex]\sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}}\\[2ex] 0.1717 = \dfrac{0.76}{\sqrt{n}}\\[2ex] n = 19.60[/tex]
Although a part of your question is missing, you might refer to this full question: Suppose the sediment density (g/cm) of a randomly selected specimen from a certain region is normally distributed with mean 2.67 and standard deviation 0.75.
(a) If a random sample of 25 specimens is selected, what is the probability that the sample average sediment density is at most 3.00? Between 2.6 and 3.00?
At most 3.00: __________
between 2.6 and 3.00: __________
(b) How large a sample size would be required to ensure that the probability in part (a) is at least 0.99?
Learn more about probability:
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