# Explain the importance of the statement "Sampling distributions play a key role in the process of statistical interference" stated by the researchers Turner and Dabney.

Question
Sampling distributions
Explain the importance of the statement "Sampling distributions play a key role in the process of statistical interference" stated by the researchers Turner and Dabney.

2021-02-26
Sampling distribution:
The probability distribution of the sample statistics when all the possible samples are drawn over the given population is termed as sampling distribution.
Justification: The sampling distribution can be defined as the probability distribution of the sample statistics that is sample mean or the sample variances that can be obtained from the samples of a given size from the population given. Therefore, it is clear that Sampling distributions play a key role in the process of statistical inference.

### Relevant Questions

Which of the following are correct general statements about the central limit theorem? Select all that apply
1. The accuracy of the approximation it provides, improves when the trial success proportion p is closer to $$50\%$$
2. It specifies the specific mean of the curve which approximates certain sampling distributions.
3. It is a special example of the particular type of theorems in mathematics, which are called Limit theorems.
4. It specifies the specific standard deviation of the curve which approximates certain sampling distributions.
5. It’s name is often abbreviated by the three capital letters CLT.
6. The accuracy of the approximation it provides, improves as the sample size n increases.
7. The word Central within its name, is mean to signify its role of central importance in the mathematics of probability and statistics.
8. It specifies the specific shape of the curve which approximates certain sampling distributions.
Which of the following are correct general statements about the Central Limit Theorem?
(Select all that apply. To be marked correct: All of the correct selections must be made, with no incorrect selections.)
Question 3 options:
Its name is often abbreviated by the three capital letters CLT.
The accuracy of the approximation it provides, improves as the sample size n increases.
The word Central within its name, is meant to signify its role of central importance in the mathematics of probability and statistics.
It is a special example of the particular type of theorems in mathematics, which are called Limit Theorems.
It specifies the specific standard deviation of the curve which approximates certain sampling distributions.
The accuracy of the approximation it provides, improves when the trial success proportion p is closer to $$50\%$$.
It specifies the specific shape of the curve which approximates certain sampling distributions.
It specifies the specific mean of the curve which approximates certain sampling distributions.
Which of the following are correct general statements about the Central Limit Theorem? Select all that apply.
1. It specifies the specific shape of the curve which approximates certain sampling distributions.
2. It’s name is often abbreviated by the three capital letters CLT
3. The word Central within its name, is meant to signify its role of central importance in the mathematics of probability and statistics.
4. The accuracy of the approximation it provides, improves when the trial success proportion p is closer to 50\%.
5. It specifies the specific mean of the curve which approximates certain sampling distributions.
6. The accuracy of the approximation it provides, improves as the sample size n increases.
7. It specifies the specific standard deviation of the curve which approximates certain sampling distributions.
8. It is a special example of the particular type of theorems in mathematics, which are called limit theorems.
Which of the following are correct general statements about the central limit theorem? Select all that apply
1. The accuracy of the approximation it provides, improves when the trial success proportion p is closer to $$50\%$$
2. It specifies the specific mean of the curve which approximates certain sampling distributions.
3. It is a special example of the particular type of theorems in mathematics, which are called Limit theorems.
4. It specifies the specific standard deviation of the curve which approximates certain sampling distributions.
5. It’s name is often abbreviated by the three capital letters CLT.
6. The accuracy of the approximation it provides, improves as the sample size n increases.
7. The word Central within its name, is mean to signify its role of central importance in the mathematics of probability and statistics.
8. It specifies the specific shape of the curve which approximates certain sampling distributions.
Explain the statement ‘The main priority with sampling distributions is to get across the idea that estimates and other statistics change every time we do a new study’.

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
Explain how to use the sampling distributions of A and B to decide which is the best estimator of $$\alpha$$.