Hypothesis Testing using z-statistic: Examples and Solutions

Introduction to Hypothesis testing using the z statistic

Introduction to Hypothesis testing using the z statistic

Remember to use σx?  in you denominator for the z statistic!!
 

1. A certain chemical pollutant in the Genesee River has been constant for several years with mean µ = 34 ppm (parts per million) and a standard deviation σ = 8 ppm. A group of factory representatives whose companies discharge liquids into the river is now claiming that they have lowered the average with improved filtration devices. Assume that their sample of size 50 gives a mean of 32.5 ppm.

State the H0 and H1 symbolically or verbally. _________________________________________

µ = ____     σ = _______ n = _______    = __________      = _______

Select a criterion (α) level and provide a reason for choosing that level. α = ________ because _____________________________________________________________________________

Is this a one-tailed or a two-tailed test? ____________________________

What is the z-value at your chosen alpha level?  ___________________

Calculate the statistic:   z = ________ 

Determine whether you should accept or reject the null hypothesis and state your conclusions verbally. 

2.  A researcher would like to determine whether there is any relationship between students’ grades and where they choose to sit in the classroom.  Specifically, the researcher suspects that the better students choose to sit in the front of the room.  To test this hypothesis, the researcher asks her colleagues to help identify a sample of n=100 students who all sit in the front row in at least one class.  At the end of the semester, the grades are obtained for these students and the average grade point average is 3.25.  For the same semester the average GPA for the entire college is µ= 2.95 with a σ= 1.10.

State the H0 and H1 symbolically or verbally. _________________________________________

µ = ____     σ = _______ n = _______    σx? = __________      = _______

Select a criterion (α) level and provide a reason for choosing that level. α = ________ because _____________________________________________________________________________

Is this a one-tailed or a two-tailed test? ____________________________

What is the z-value at your chosen alpha level?  ___________________

Calculate the statistic:   z = ________ 

Determine whether you should accept or reject the null hypothesis and state your conclusions verbally. 

3. Fun size packs of Halloween M & M’s have been purported to contain 15 candy pieces per package with a standard deviation of 1.5 pieces. You collect all your daughter’s M & M candies she received on Halloween – you open them up and count them (much to your daughter’s dismay). You find that the 16 packages she has have a mean of 13.8 pieces. Do you think the M & M Company is taking advantage of the Halloween season by putting fewer candies in each package?

State the H0 and H1 symbolically or verbally. _________________________________________

µ = ____     σ = _______ n = _______    σx? = __________      = _______

Select a criterion (α) level and provide a reason for choosing that level. α = ________ because _____________________________________________________________________________

Is this a one-tailed or a two-tailed test? ____________________________

What is the z-value at your chosen alpha level?  ___________________

Calculate the statistic:   z = ________ 

Determine whether you should accept or reject the null hypothesis and state your conclusions verbally. 

4. A researcher is evaluating the effectiveness of a new physical education program for elementary school children. The program is designed to reduce competition and increase individual self-esteem. A sample of n = 16 children is selected and the children are placed in the new program. After 3 months, each child is given a standardized self-esteem test. For the general population of elementary school children, the scores on the self-esteem test form a normal distribution with µ = 40 and a σ = 8.

a. If the researcher obtains a sample mean of M = 42, is this enough evidence to conclude that the program has a significant effect? Assume a two tailed test with α = .05.

b. If the sample mean is 44, is this enough to demonstrate a significant effect? Again, assume a two tailed test with α = .05.

c. Briefly explain why you reach different conclusions for part (a) and part (b).

d. What would you conclude if you increased the sample size to 25 for both (a) and (b)?