Improve Your Understanding of College Level Statistics Problems

A population of values has a normal distribution with ${\displaystyle \mu =200.7}$ and ${\displaystyle \sigma =10}$. You intend to draw a random sample of size ${\displaystyle n=178}$
Find the probability that a sample of size ${\displaystyle n=178}$ is randomly selected with a mean less than 198.6.
${\displaystyle P\left(M<198.6\right)=}$?

A distribution of values is normal with a mean of 79.5 and a standard deviation of 7.4.
Find the probability that a randomly selected value is between 71.4 and 78.
${\displaystyle P\left(71.4<x<78\right)=P\left(<z<\right)=}$?

Which of the following is not a condition for performing inference about a population mean U ? 1) Inference is based on n independent measurements. 2) The population distribution is Normal or the sample size is large (say n > 30). 3) The data are obtained from a SRS from the population of interest. 4) The population standard deviation, ?

Use the table from the Theoretical Distribution section to calculate the following answers. Round your answers to four decimal places. $P(x=3)=?$
$P(1<x<4)=?$
$P(x\ge 8)=?$ Use the data from the Organize the Data section to calculate the following answers. Round your answers to four decimal places. $RF(x=3)=?$
$RF(1<x<4)=?$
$RF(x\ge 8)=?$ Discussion Questions 1. Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical, empirical, and simulation distributions. Use complete sentences.

The random variable X follows a normal distribution ${\displaystyle ?(20,102)}$.
Find ${\displaystyle P(10<?<35)}$,

The article “Anodic Fenton Treatment of Treflan MTF” describes a two-factor experiment designed to study the sorption of the herbicide trifluralin. The factors are the initial trifluralin concentration and the \(\displaystyle{F}{e}^{{{2}}}\ :\ {H}_{{{2}}}\ {O}_{{{2}}}\) delivery ratio. There were three replications for each treatment. The results presented in the following table are consistent with the means and standard deviations reported in the article. \(\begin{array}{|c|c|}\hline \text{Initial Concentration (M)} & \text{Delivery Ratio} & \text{Sorption (%)} \\ \hline 15 & 1:0 & 10.90 \quad 8.47 \quad 12.43 \\ \hline 15 & 1:1 & 3.33 \quad 2.40 \quad 2.67 \\ \hline 15 & 1:5 & 0.79 \quad 0.76 \quad 0.84 \\ \hline 15 & 1:10 & 0.54 \quad 0.69 \quad 0.57 \\ \hline 40 & 1:0 & 6.84 \quad 7.68 \quad 6.79 \\ \hline 40 & 1:1 & 1.72 \quad 1.55 \quad 1.82 \\ \hline 40 & 1:5 & 0.68 \quad 0.83 \quad 0.89 \\ \hline 40 & 1:10 & 0.58 \quad 1.13 \quad 1.28 \\ \hline 100 & 1:0 & 6.61 \quad 6.66 \quad 7.43 \\ \hline 100 & 1:1 & 1.25 \quad 1.46 \quad 1.49 \\ \hline 100 & 1:5 & 1.17 \quad 1.27 \quad 1.16 \\ \hline 100 & 1:10 & 0.93 \quad 0.67 \quad 0.80\\ \hline \end{array}\) a) Estimate all main effects and interactions. b) Construct an ANOVA table. You may give ranges for the P-values. c) Is the additive model plausible? Provide the value of the test statistic, its null distribution, and the P-value.

For each set of data below, draw a scatterplot and decide whether or not the data exhibits approximately periodic behaviour.

a) $\begin{array}{|cccccccccccccc|}\hline x& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ y& 0& 1& 1.4& 1& 0& -1& -1.4& -1& 0& 1& 1.4& 1& 0\\ \hline\end{array}$

b) $\begin{array}{|cccccc|}\hline x& 0& 1& 2& 3& 4\\ y& 4& 1& 0& 1& 4\\ \hline\end{array}$

c) $\begin{array}{|ccccccccc|}\hline x& 0& 0.5& 1.0& 1.5& 2.0& 2.5& 3.0& 3.5\\ y& 0& 1.9& 3.5& 4.5& 4.7& 4.3& 3.4& 2.4\\ \hline\end{array}$

d) $\begin{array}{|cccccccccccc|}\hline x& 0& 2& 3& 4& 5& 6& 7& 8& 9& 10& 12\\ y& 0& 4.7& 3.4& 1.7& 2.1& 5.2& 8.9& 10.9& 10.2& 8.4& 10.4\\ \hline\end{array}$

A kickball's height h (in feet) t seconds after it is kicked off the ground is represented by the function ${\displaystyle h=-16t^2+48t}$ (in math terms). Calculate the kickball's highest point. b) Locate and examine the symmetry axis. 

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)?