Respuesta :
Answer:
a) The sampling distirbution has a mean of 135 and a standard deviation of 1.732.
b) The sampling distribution is also bell shaped, centered in the population mean. It has less spread that the population distribution because the standard deviation is smaller.
c) P(X>140)=0.002
Step-by-step explanation:
a) We are taking samples of size n=3, as we are taking 3 readings daily.
Each reading has a mean of 135 and a standard deviation of 3.
The sampling distribution depends on the population parameters and the sample size.
The mean of the sampling distribution is equal to the population mean. In this case, the mean of the sampling distribution is 135.
The standard deviation depends on the standard deviation of the population and the sample size, and is calculated as:
[tex]\sigma_M=\dfrac{\sigma}{\sqrt{N}}=\dfrac{3}{\sqrt{3}}=1.732[/tex]
b) The sampling distribution is also bell shaped, centered in the population mean. It has less spread that the population distribution because the standard deviation is smaller.
c) The probability that the sample mean exceeds 140 can be calculated using the standard normal distribution.
First, we have to calculate the z-score for X=140. We use for this the mean and standard deviation of the sampling distribution.
[tex]z=\dfrac{X-\mu_M}{\sigma_M}=\dfrac{140-135}{1.732}=\dfrac{5}{1.732}=2.887[/tex]
With this z-score, we can look up the probabilty in the standard normal distribution
[tex]P(X>140)=P(z>2.887)=0.002[/tex]
(a): The required mean and standard deviations are 135 and 1.134
(b): The shape is normal
(c): The required probability is 0.
Z-score:
It indicates how many standard deviations an entity is, from the mean.
Given that,
mean([tex]\mu[/tex])=135
standard deviation ([tex]\sigma[/tex])=3
[tex]n = 7[/tex]
Part(a):
[tex]\mu_{\bar{x}}=135[/tex]
[tex]\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n} } \\=\frac{3}{\sqrt{7} } \\=1.134[/tex]
Part(b):
The shape of sampling distribution is normal.
Part(c):
[tex]P(\bar{x} > 140)=1-P(\bar{x} < 140)\\=1-P(z < 4.41)\\=1-1\ [by\ z\ score]\\=0[/tex]
Learn more about the topic Z-score:
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