Confidence Intervals & Hypothesis Tests: Z-Test Function & Application to 3 Populations, Study Guides, Projects, Research of Mathematics

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Project 6

Confidence Intervals and Tests of hypotheses

Problem 1. We begin this project by writing an R function which we call z.test that inputs a set of data and performs a z test for the mean and produces a two-sided confidence interval. Specifically, z.test will have the following input and output.

Input: A vector (call it x ) containing the data A string variable (call it alternative ) that inputs the form of the alternative: ‘g’ for , ‘l’ for , and ‘t’ for two sided tests with. [Default: alternative = ‘t’ ] A variable (call it mu ) containing the value of in the null hypothesis. [Default: mu = 0 ] A variable (call it conf.level ) that gives the confidence level of the confidence interval desired. [Default: conf.level = 0.95 ]

Output: Statistic : the value of the z statistic Pvalue: the p-value for the test Lconf: the lower limit of the confidence interval Uconf: the upper limit of the confidence interval A normal quantile plot of the data

Consider the following three (population) sources from which we take samples:

1. Data Generated from a normal with mean 10 and standard deviation 2.
2. Data generated from Binomial with n = 5, and p= 0.1.
3. Quebec Birth Data: Number of daily births in Quebec from January 1, 1977 to December 31, 1990.

Problem 2: For each of the above populations perform the following steps: a) Obtain the population mean, call it. b) Take a sample of size 5 from the population and test the hypothesis versus. Repeat this 5000 times, and determine the proportion of times that you reject the null hypothesis at 5% level (this is called the observed significance level). Compare this value to the 5% level (called the nominal significance level) and comment, c) Repeat (b) using 10% significance level. d) Compute s 99% confidence interval for and determine what proportion of the intervals contain the population mean Comment on the proportions that you obtain.

Problem 3: Repeat problem 2 with the modification that in part (b), take a sample of size 50 rather than a sample of size 5.