Question

1. A researcher wanted to estimate the mean contributions made to charitable causes by all major companies. A random sample of 18 companies produced by the following data on contributions (in millions of dollars) made by them.
1.8, 0.6, 1.2, 0.3, 2.6, 1.9, 3.4, 2.6, 0.2
2.4, 1.4, 2.5, 3.1, 0.9, 1.2, 2.0, 0.8, 1.1

Assume that the contributions made to charitable causes by all major companies have a normal distribution.
a. What is the point estimate for the population mean?
b. Construct a 98% confidence interval for the population mean.
c. What sample size would the researcher need to obtain a margin of error of 100,000 for the same confidence level? (Assume that the sample standard deviation obtained from his original sample is equal to the population standard deviation.)
d. Prior to collecting the data, the researcher believed that the mean contribution of all companies was less than $2.5 million. For a significance level of 0.01, test the researchers hypothesis.

Answer 1

(a)

From the given data, the following statistics are calculated:

n = 18

= 1.6667

s = 0.9292

So,

Point Estimate of the Population Mean = Sample Mean = 1.67

So,

Answer is:

1.67

(b)
SE = s/

= 0.9292/

= 0.2190

= 0.02

ndf = n - 1 = 18 - 1 = 17

From Table, critical values of t = 2.5669

Confidence Interval:

1.6667 (2.5669 X 0.2190)

= 1.6667 0.5622

= ( 1.10 ,2.23)

Confidence interval:

1.10 < < 2.23

(c)

Sample Size (n) is given by:

Given = 0.02

From Table, critical values of Z = 2.33

= 0.9292

e = 0.1

Substituting, we get:

So,

Answer is:

469

(d)

H0 : Null Hypothesis:

HA: Alternative Hypothesis:

SE = s/

= 0.9292/

= 0.2190

Test Statistic is given by:
t = (1.6667 - 2.5)/0.2190

= - 3.8050

= 0.01

ndf = 17

From Table, critical value of t = - 2.5669

Since calculated value of t = - 3.8050 is less than critical value of t = - 2.5569, the difference is significant. Reject null hypothesis.

Conclusion:
The data support the claim that the mean contribution of all companies was less than $2.5 million.