Derive 100(1−α) percent lower and upper confidence intervals for p, when the data consist of the values of n independent Bernoulli random variables wi

Assume that when adults with smartphones are randomly selected, 57​% use them in meetings or classes. If 5 adult smartphone users are randomly selected, find the probability that at least 3 of them use their smartphones in meetings or classes.
 

A balanced experimental design has a sample size of $n=11$ observations at each of $k=3$ factor levels. The sample averages are ${\bar{x}}_{1\cdot }=48.05,{\bar{x}}_{2\cdot }=44.74,and{\bar{x}}_{3\cdot }=49.11,andMSE=4.96$.
(a) Calculate pairwise confidence intervals for the factor level means with an overall confidence level of 95%.
(b) Make a diagram showing which factor level means are known to be different and which ones are indistinguishable.
(c) What additional sampling would you recommend to reduce the lengths of the pairwise confidence intervals to no more than 2.0 ?

An analysis of variance of the yields of five different varieties of wheat, observed on one plot each at each of six different locations. The data from this randomized block design are listed here:
$\begin{array}{|ccccccc|}\hline Varieties& Location1& Location2& Location3& Location4& Location5& Location6\\ A& 35.3& 31.0& 32.7& 36.8& 37.2& 33.1\\ B& 30.7& 32.2& 31.4& 31.7& 35.0& 32.7\\ C& 38.2& 33.4& 33.6& 37.1& 37.3& 38.2\\ D& 34.9& 36.1& 35.2& 38.3& 40.2& 36.0\\ E& 32.4& 28.9& 29.2& 30.7& 33.9& 32.1\\ \hline\end{array}$
a. Use the appropriate nonparametric test to determine whether the data provide sufficient evidence to indicate a difference in the yields for the five different varieties of wheat. Test using $\alpha \alpha =.05$.
b. How do the analysis of variance F test compare with the test in part a? Explain.

Sharon draws a random card, from a regular deck of cards, and rolls a regular 6 sided die. What is the probability that Sharon draws a card that is a Heart and rolls a 3?

Find the probability density function of $Y=e^X$, when X is normally distributed with parameters $\mu \text{and}{\sigma }^2$. The random variable Y is said to have a lognormal distribution (since log Y has a normal distribution) with parameters $\mu \text{and}{\sigma }^2$