## EFR515 Hypothesis testing课程简介

This course introduces the logic and methods of statistics. We begin with a discussion of the role of statistics, introducing the concepts of internal and external validity.
Common methods for describing the characteristics of individuals and educational outcomes are presented, including the use of graphs and summary measures such as the mean, median, standard deviation, and correlation coefficient. Because of the large natural variability among individuals, one must be able to determine whether or not an apparent difference or patterns present in the data seems to be merely a chance occurrence. Probability concepts are introduced to help us in this effort.

## PREREQUISITES

Probability concepts are introduced to help us in this effort. Probability then forms the basis of all of the inferential statistical procedures subsequently presented. At the end of the course, students will:

• identify basic statistical applications for educational research.
• explain how to implement quantitative approaches to educational research.
• identify types of statistical methods and strategies, and select data collection and analysis approaches for different research interests.
• integrate fundamental statistical theories and concepts with functions of SPSS programs in the context of an analysis project’s overall design.

## EFR515 Hypothesis testing HELP（EXAM HELP， ONLINE TUTOR）

1. During the $1980 \mathrm{~s}$, the general consensus is that about $5 \%$ of the nation’s children had autism. Some claimed that increases certain chemicals in the environment has led to an increase in autism.
(a) Write an appropriate hypothesis test for this situation.
(b) Give an appropriate test for this hypothesis, stating what are the necessary conditions for performing the test.
(c) A recent study examined 384 children and found that 46 showed signs of autism. Perform a test of the hypothesis and state the $p$-value.
(d) What are your conclusions? State how you use the $p$-value.

a) Hypothesis test: $H_0$: The proportion of children with autism is equal to or less than $0.05$ $H_1$: The proportion of children with autism is greater than $0.05$

b) The appropriate test for this hypothesis is a one-tailed test of proportions, using a normal approximation. The necessary conditions for performing this test are that the sample is a random sample, the sample size is large enough (i.e., np≥10 and n(1-p)≥10), and the observations are independent.

c) The sample proportion of children with autism is $\hat{p} = \frac{46}{384} \approx 0.120$. The test statistic is:

$$z = \frac{\hat{p} – p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{0.120-0.05}{\sqrt{\frac{0.05(1-0.05)}{384}}} \approx 5.08$$

Using a standard normal table or calculator, the $p$-value is very small, approximately $p<0.00001$.

d) Since the $p$-value is less than the significance level (not specified in the question), we reject the null hypothesis. There is evidence to suggest that the proportion of children with autism is greater than $0.05$. This means that the claim that certain chemicals in the environment are causing an increase in autism is supported by this data. However, it is important to note that this study alone cannot prove causation and further research is necessary. The $p$-value is used as a measure of the strength of evidence against the null hypothesis, and the smaller it is, the stronger the evidence against the null hypothesis.

1. A company with a fleet of 150 cars found that the emission system of 7 our of the 22 cars tested failed to meet pollution guidelines.
(a) Write a hypothesis to test if more than $20 \%$ of the entire fleet might be out of compliance.
(b) Test the hypothesis based on the binomial distribution and report a $p$-value.
(c) Is the test significant at the $10 \%, 5 \%, 1 \%$ level?

a) Hypothesis test: $H_0$: The proportion of cars out of compliance is less than or equal to $0.20$ $H_1$: The proportion of cars out of compliance is greater than $0.20$

b) We can use the binomial distribution to model the number of cars out of compliance. The probability of a car being out of compliance is $p=\frac{7}{22} \approx 0.318$. Let $X$ be the number of cars out of compliance in a sample of 150 cars. Then, under the null hypothesis, $X$ follows a binomial distribution with $n=150$ and $p_0=0.20$. The test statistic is:

$$z = \frac{\hat{p} – p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{0.318-0.20}{\sqrt{\frac{0.20(1-0.20)}{150}}} \approx 4.07$$

Using a standard normal table or calculator, the $p$-value is very small, approximately $p<0.00001$.

c) The test is significant at the $1 %$ level, since the $p$-value is less than $0.01$. It is also significant at the $5 %$ and $10 %$ level, since the $p$-value is less than $0.05$ and $0.10$, respectively.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.