Class Activity 18 - Review for Review

Topics to know:

calculating confidence intervals for a difference in proportions (don’t use a pooled proportion)
doing a hypothesis test for a difference in proportions (null hypothesis is usually that the proportions are equal so use a pooled proportion)
calculating confidence intervals for a difference in means
doing a hypothesis test for a difference in means (null hypothesis is usually that the means are equal)
Understand what a $p$ -value is and how to use it to form conclusions
Understand decision errors: type I and type II errors
know how to identify paired data
Be able to perform hypothesis tests using both $z$ -tests and $t$ -tests (proportions are always a $z$ -test, means are a $z$ -test when you know the population standard deviation and a $t$ -test when you only know the sample standard deviation)
Understand how to compute power and how to interpret power and decision errors in graphs

Class Acticity 17 is copied here for your convenience

The local electric company says that customers pay less than $\$200$ per month on average. A local reporter is suspicious of these claims and thinks that customers are spending more than $\$ 200$ per month. A random sample of $35$ customers bills during a given month is taken with a mean of $220$ and a standard deviation of $45$ . Create a hypothesis test from the perspective of the reporter to test the claims of the company under a $5 \%$ significance level.
Compute the power of the hypothesis test at detecting a mean of $\$220$ . (Make sure to draw a picture)
A population of people has incomes with a standard deviation of $\sigma = \$9000$ . A random sample of $25$ people had a mean income of $\$54,800$ . Perform a hypothesis test using a significance level of $.05$ to see if this data provides statistically significant evidence that the mean salary is different than $\$60,000$ .
What is the power of the test at detecting a mean income of $ $ 57000$?

Given two proportions $\hat{p}_1 = .25$ and $\hat{p}_2=.30$ from sample size $n_1 = 35$ and $n_2=55$ . Calculate a $90 \%$ confidence interval.
Given two proportions $\hat{p}_1 = .25$ and $\hat{p}_2=.30$ from sample size $n_1 = 35$ and $n_2=55$ . Perfrom a hypothesis test to see if the data provides evidence that there is a difference int the two proportions.
True/False: A large $t$ -score corresponds to a large $p$ -value
True/False: In a one tailed test we compute a $t$ -score of $1.7$ . With a $10 \%$ significance level we fail to reject the null hypothesis.
Describe a Type I error in the context of problem one
1. Describe a Type II error in the context of problem one
In a random sample of $1000$ students, $\hat{p} = 0.8$ (or $80\%$ ) were in favor of longer hours at the school library. Find the standard error of $\hat{p}$ .
When should you use a pooled proportion?
Null and alternate hypothese are statements about:

Consider a random sample of $100$ females and $100$ males. Suppose $15$ of the females are left-handed and $12$ of the males are left-handed. What is the estimated difference between population proportions of females and males who are left-handed? Select the choice with the correct notation and numerical value.