Understanding Sampling Distribution with Plainmath

The following data represent soil water content for independent random samples of soil taken from two experimental fields growing bell peppers Soil water content from field I: $x_1;n_1$=72 15.111.210.310.816.68.39.112.39.114.3 10.716.110.215.28.99.59.611.31411.3 15.611.213.89.08.48.21213.911.616 9.611.48.48.014.110.913.213.814.610.2 11.513.114.712.510.211.811.012.710.310.8 11.012.610.89.611.510.611.710.19.79.7 11.29.810.311.99.711.310.4121110.7 8.811.1 Soil water content from field II: $x_2;n_2$= 80 12.110.213.68.113.57.811.87.78.19.2 14.18.913.97.512.67.314.912.27.68.9 13.98.413.47.112.47.69.9267.37.4 14.38.413.27.311.37.59.712.36.97.6 13.87.513.38.011.36.87.411.711.87.7 12.67.713.213.910.412.87.610.710.710.9 12.511.310.713.28.912.97.79.79.711.4 11.913.49.213.48.811.97.18.51414.2
Which distribution (standard normal or Student's t) did you use? Why? Do you need information about the soil water content distributions?

1) Describe sampling distributions and sampling variavility
2) Explain The Central Limit Theorem
3) Explain how confidence intervals are created and what can they tell us about population parameters

The correct statement which is incorrect from the options about the sampling distribution of the sample mean
(a) the standard deviation of the sampling distribution will decrease as the sample size increases,
(b) the standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples,
(c) the sample mean is an unbiased estimator of the true population mean,
(d) the sampling distribution shows how the sample mean will vary in repeated samples,
(e) the sampling distributions shows how the sample was distributed around the sample mean.

Which of the following is true about the sampling distribution of means?
A. Shape of the sampling distribution of means is always the same shape as the population distribution, no matter what the sample size is.
B. Sampling distributions of means are always nearly normal.
C. Sampling distributions of means get closer to normality as the sample size increases.
D. Sampling distribution of the mean is always right skewed since means cannot be smaller than 0.

Which of the following statements about the sampling distribution of the sample mean is incorrect?
(a) The standard deviation of the sampling distribution will decrease as the sample size increases.
(b) The standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples.
(c) The sample mean is an unbiased estimator of the population mean.
(d) The sampling distribution shows how the sample mean will vary in repeated samples.
(e) The sampling distribution shows how the sample was distributed around the sample mean.

Explain whether the central limit theorem can be applied and assert that the sampling distributions of A and Bare approximately normal, if the sample sizes of A and Bare large.

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.

Explain the meaning of the phrase " the sampling distribution of the sample statistic A"

How can relative frequencies be used yo help us estimate probabilities occuring in sampling distributions?

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 $50\mathrm{\%}$
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 is true about sampling distributions?
-Shape of the sampling distribution is always the same shape as the population distribution, no matter what the sample size is.
-Sampling distributions are always nearly normal.
-Sampling distribution of the mean is always right skewed since means cannot be smaller than 0.
-Sampling distributions get closer to normality as the sample size increases.

Whether the provided statement is true or false.

To explain: The ways in which relative frequencies can be used to help us in estimating probabilities occurring in 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 $50\mathrm{\%}$.
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.

Obtain the sampling distribution of $\bar{p}$

Suppose that ${\displaystyle Y_1,Y_2,Y_3}$ denote a random sample from anexponential distribution with density function ${\displaystyle f\left(y\right)=\left(\frac{1}{\lambda }\right)\cdot e^{\frac{-y}{\theta }},y>0}$
=0 elsewhere
Consider the following five estimators of ${\displaystyle \theta }$:
${\displaystyle {\hat{\theta }}_1=Y_1,{\hat{\theta }}_2=\frac{Y_1+Y_2}{2},{\hat{\theta }}_3=\frac{Y_1+2Y_2}{3},{\hat{\theta }}_4=\bar{Y}}$
a) Which of these estimators are unbiased?
b) Among the unbiased estimators, which has the smallest variance?

Critical Thinking Let x be a random variable representing the amount of sleep each adult in New York City got last night. Consider a sampling distribution of sample means $\bar{x}$.
What value will the standard deviation ${\sigma }_{\bar{x}}$ of the sampling distribution approach?

To explain:Whether the given statement is correct or not

Explain the meaning of assertion that A is an unbiased estimator of $\alpha$

The Wall Street Journal reported that the age at first startup for $55\mathrm{\%}$ of entrepreneurs was 29 years of age or less and the age at first startup for $45\mathrm{\%}$ of entrepreneurs was 30 years of age or more.
a. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is the sample proportion of entrepreneurs whose first startup was at 29 years of age or less.
b. Suppose a sample of 200 entrepreneurs will be taken to learn about the most important qualities of entrepreneurs. Show the sampling distribution of $\bar{p}$ where $\bar{p}$ is now the sample proportion of entrepreneurs whose first startup was at 30 years of age or more.
c. Are the standard errors of the sampling distributions different in parts (a) and (b)?