Respuesta :
Options :
a.) shape unknown with mean of 2.8 and a standard deviation of 0.4 b.) approximately normal with mean of 2.8 and standard deviation of 4 C.) shape unknown with mean of 2.8 and standard deviation of 4 d.) approximately normal with mean of 2.8 and standard deviation of 0.4
Answer: d.) approximately normal with mean of 2.8 and standard deviation of 0.4.
Step-by-step explanation:
Given that :
Population Mean (m) = 2.8
Population Standard deviation (s) = 4
Sample size = 100
The mean of the sampling distribution is equal to the population mean, according to the central limit theorem. As the size of the sample increases, the mean if the sample distribution converges to that if the population mean.
Hence, sample mean = 2.8
The sample standard deviation (standard error) :
Standard Error = population standard deviation / √sample size
= s/√n
= 4 / √100
= 4/10
= 0.4
Using the Central Limit Theorem, it is found that the correct description of the sampling distribution of the sample means is:
Approximately normal, with mean of 2.8 and standard deviation of 0.4.
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
- For a variable of unknown distribution, the Central Limit Theorem can be applied if the sample size is larger than 30.
In this problem, the population has an unknown distribution, with mean [tex]\mu = 2.8[/tex] and standard deviation [tex]\sigma = 4[/tex]
The sampling distribution of the sample means of size 100 has:
- Approximately normal shape, as the sample size is [tex]n = 100 > 30[/tex].
- Mean [tex]\mu = 2.8[/tex]
- Standard deviation [tex]s = \frac{4}{\sqrt{100}} = 0.4[/tex].
A similar problem is given at https://brainly.com/question/14099217
