ECI 114 Statistics Final Exam Winter 2020 - Problems and Solutions

This document presents the solutions to the ECI 114 Winter 2020 final exam in statistics. The exam covers a range of statistical concepts, including hypothesis testing, confidence intervals, and power analysis. The solution begins with a true/false section assessing understanding of fundamental statistical principles. Following this, the document provides a detailed solution to a problem involving hypothesis testing related to manufacturing flaws in football helmets. This includes calculating the test statistic, making a conclusion based on the p-value, constructing a confidence interval, and calculating the power of the test and the required sample size to achieve a specific power level. All steps are shown with final answers highlighted.

1.
2. For each of the following statements, circle the letter “T” if it is true, and “F” if it is false.
(2 *9 = 18)
T F (a) The number of degrees of freedom in the numerator of the f statistic is n1-1 when
testing σ1 vs. σ2.
T
T F (b) In regre ssion anal ysi s, the term Sxx quantifie s the scatter of x about 𝑥𝑥̅ .
F
T F (c) Zα < Tα, v for all cases α<0.5 when v < 40.
T
T F (d) The standard error for the population proportion derived from sample data is
p(1-p)/n
F
T F (e) Since the definition of population 1 and 2 are interchangeable when conducting
hypothesis testing on the variance, fα,u,v = fα,v,u for all α, u, v.
F
T F (f) 2α1,v > 2α2,v for all values of α1<0.5 and α2<0.5 as long as α1 > α2.
F
T F (g) √ s2
n i s the standard error of the sample variance.
T
T F (h) The p-value defines the acceptance region of a hypothesis test.
T
T F (i) Linear least squares regression analysis assumes that the independent variable x and
the dependent variable y are normally distributed.
T
1. (20 pts) A researcher claims that more than 10% of football helmets have manufacturing
flaws that could potentially cause injury to the wearer. A sample of 85 helmets revealed
that 9 helmets contained such defects.
(a) (5 pts) Perform a hypothesis test to determine if the data supports the researchers
claim. Use α = 0.05.
p0=0.1
p= 9
85 =0.1059
H0 : p= p0
H1 : p> p0

Z= p−p0
√ p0 ( 1− p0 )
n
= 0.1059−0.1
√ 0.1 ( 1−0.1 )
85
=0.181
Z(0.05) = 1.645
As evident, the z calculated 0.181 is less than critical value 1.645 thus we fail to reject
the null hypothesis and conclude that manufacturing flaws of the football helmets is
not more than 10%.
(b) (5 pts) Construct a confidence interval to determine if the data supports the researchers
claim. Use α = 0.05.
CI = p± Z α
2 √ p0 ( 1− p0 )
n
CI =0.1059 ±1.96 √ 0.1 ( 1−0.1 )
85 =0.04∧0
(c) (5 pts) What is the power of this test if the true defect rate is as high as 11%?
Z= p−p0
√ p0 ( 1− p0 )
n
=1.6 45= p−0. 1
√ 0.1 ( 1−0.1 )
85
0.0 5 3 5= p−0.1
p=0.0535+ 0.1=0.1535
z= 0.1535−0.1 1
√ 0. 11 ( 1−0.1 1 )
85
=1.282
P(1.282) = 0.90
Thus, the power of the test is 90%
(d) (5 pts) Estimated the sample size required to achieve a power of 95% if the true defect
rate is as high as 11%?
z=1.645= 0.1535−0.11
√ 0.11 ( 1−0.11 )
n
= 0.0979
n =0.000699
n=140