Improve Your Understanding of College Level Statistics Problems

Consider a random sample of size n = 31, with sample mean ${\displaystyle \bar{x}=45.2}$ and sample standard deviation s = 5.3. Compute 90%, 95%, and 99% confidence intervals for \muμ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $$ ${\displaystyle lim\{n\to \mathrm{\infty }\}\frac{a\left\{2n\right\}}{a_n}<\frac{1}{2}}$ then $\sum a_n$converges,while if ${\displaystyle lim\{n\to \mathrm{\infty }\}\frac{a\{2n+1\}}{a_n}>\frac{1}{2}}$, then $\sum a_n$ diverges.

Let ${\displaystyle a_n=\frac{1}{1+x}\frac{2}{2+x}\dots \frac{n}{n+x}\frac{1}{n}=\frac{(n-1)!}{(1+x)(2+x)\dots (n+x)}}$.

Show that $\frac{a_{2n}}{a_n}\le \frac{e^{-x/2}}{2}$ . For which x > 0 does the generalized ratio test imply convergence of $\sum _{n=1}^{\mathrm{\infty }}{a}_n$?

B. G. Cosmos, a scientist, believes that the probability is ${\displaystyle \frac{2}{5}}$ that aliens from an advanced civilization on Planet X are trying to communicate with us by sending high-frequency signals to Earth. By using sophisticated equipment, Cosmos hopes to pick up these signals. The manufacturer of the equipment, Trekee, Inc., claims that if aliens are indeed sending signals, the probability that the equipment will detect them is ${\displaystyle \frac{3}{5}}$. However, if aliens are not sending signals, the probability that the equipment will seem to detect such signals is $\frac{1}{10}$. If the equipment detects signals, what is the probability that aliens are actually sending them?

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.
$H_0:{\mu }_1={\mu }_2.{\mu }_1>{\mu }_2$
$H_0:{\mu }_1<{\mu }_2.{\mu }_1={\mu }_2$
$H_0:{\mu }_1={\mu }_2.{\mu }_1<{\mu }_2$
$H_0:{\mu }_1={\mu }_2.{\mu }_1\ne {\mu }_2$
(b) What sampling distribution will you use? What assumptions are you making?

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.)

(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
${\mu }_1-{\mu }_2$.
(Round your answers to two decimal places.)
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.

The two-way table summarizes data on whether students at a certain high school eat regularly in the school cafeteria by grade level.

$\text{Grade}\text{Eat in cafeteria}\begin{array}{lrrrrr}& 9\mathrm{th}& 10\mathrm{th}& 11\mathrm{th}& 12\mathrm{th}& \text{ Total }\\ \text{ Yes }& 130& 175& 122& 68& 495\\ \text{ No }& 18& 34& 88& 170& 310\\ \text{ Total }& 148& 209& 210& 238& 805\end{array}$

If you choose a student at random who eats regularly in the cafeteria, what is the probability that the student is a 10th-grader?

A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-50s.

$\begin{array}{lc}& \text{Soccer experience}\\ \text{Arthritis}& \begin{array}{lcccc}& & & \text{ Did not }& \\ & \text{ Elite }& \text{ Non-elite }& \text{ play }& \text{ Total }\\ \text{ Yes }& 10& 9& 24& 43\\ \text{ No }& 61& 206& 548& 815\\ \text{ Total }& 71& 215& 572& 858\end{array}\end{array}$

Suppose we choose one of these players at random. What is the probability that the player has arthritis?

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?

A study in Sweden looked at former elite soccer players, people who had played soccer but not at the elite level, and people of the same age who did not play soccer. Here is a two-way table that classifies these individuals by whether or not they had arthritis of the hip or knee by their mid-fifties: $$\begin{gathered} \text{Soccer level} \\ \begin{array}{llccc}& & \text{ Ellte }& \text{ Non-elite }& \text{ Did not play }\\ \text{ Whether person }& \text{ Yes }& 10& 9& 24\\ \text{ developed arthritis }& \text{ No }& 61& 206& 548\end{array} \end{gathered}$$ Researchers suspected that the more serious soccer players were more likely to develop arthritis later in life. Do the data confirm this suspicion? Calculate appropriate percentages to support your answer.