Respuesta :
Answer:
a) Mean = 133, Standard deviation = 2.5
b) Normal distribution
c) 0.0026
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 4
Mean = 133
Standard deviation = 5
a) Mean and the standard deviation of the sampling distribution of the sample mean for the four observations each day.
The sampling distribution has a mean equal to the population mean
[tex]\bar{x} = \mu = 133[/tex]
The standard deviation of the sampling distribution is given by:
[tex]\sigma_s = \dfrac{\sigma}{\sqrt{n}} = \dfrac{5}{\sqrt{4}} = 2.5[/tex]
b) Shape of the sampling distribution
Central limit theorem:
When a large sample is drawn from a normal distribution, then the distribution of sample means approaches a normal distribution.
Since the distribution is normal, the sampling distribution of the sample mean will be bell shaped and follow a normal distribution.
c) P(sample mean exceeds 140)
[tex]P( x > 140) = P( z > \displaystyle\frac{140 - 133}{2.5}) = P(z > 2.8)[/tex]
[tex]= 1 - P(z \leq 2.8)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x > 140) = 1 - 0.9974 = 0.0026[/tex]
0.0026 is the probability that the sample mean exceeds 140.
