Download partial solutions to test two sample questions.

ELEMENTARY STATISTICS
Page 1 of 2.
SAMPLE QUESTIONS SET TWO.
SHOW ALL CALCULATIONS/TI COMMANDS. GIVE ANSWERS TO AT LEAST SIX DECIMAL PLACES.
1. In a random sample of 80 cars, 14 have a satellite radio receiver. Find the 95% confidence
interval for the percentage of cars with such a receiver. Show calculation of margin of error.
p = 14/80 = 0.175, q = 66/80 = 0.825,  = 0.05, /2 = 0.025.
7 33
40  40
E = z(0.025)
= invNorm(0.975 ) 231/40^2/80 = 0.0832624329.
80
Store this in variable E.
Compute p − E and p + E.
The 95% confidence interval is 0.0917375671 < p < 0.2582624329.
2. If the sample mean is 43 for a sample of size 120, and the population standard deviation is
6.25, find the 94% confidence interval for the population mean. Give the endpoints
accurate to six or more decimal places. Show the calculation of the margin of error.
n = 120, x = 43,  = 6.25,  = 0.06, /2 = 0.03.
6.25
E = z(0.03 ) 
= invNorm(0.97 )  6.25  120 = 1.073076131.
120
Store this in variable E.
Compute x − E and x + E.
The 94% confidence interval is 41.92692387 <  < 44.07307613.
3. Using a sample of size n, the endpoints of a confidence interval for the population mean are
given by x  1.43953( / n ).
(a) What is the value of  ? (Can round to three decimal places.)
z(/2 ) = 1.43953. /2 = P(z  1.43953 )  normalcdf(1.43953, 9 ) 
0.0750002515.
  0.1500005031. Or, about 15.0%.
(b) What is the confidence level? c = 1 − . About 85.0%.
(c) If  is 7.0, find the minimum sample size so that the margin of error, E, is
under 0.65 in size.
n 
z(/2 ) 2
E
=
1.43953(7.0 ) 2
= 240.3315608
0.65
The minimum sample size is 241.
GO TO PAGE 2.
ELEMENTARY STATISTICS
SAMPLE QUESTIONS SET TWO.
Page 2 of 2.
SHOW ALL CALCULATIONS/TI COMMANDS. GIVE ANSWERS TO AT LEAST SIX DECIMAL PLACES.
4. If the sample mean is 41, for a sample of size 14, and the sample standard deviation is 4.5,
find the 97% confidence interval for the population mean. (The population is normally
distributed. Show calculation of maximum error.)
n = 14, df = n − 1 = 13, x = 41, s = 4.5,  = 0.03, /2 = 0.015.
t(0.015, 13 ) = invT(0.985, 13 ) = 2.435845205.
Or, solve 0 = 0.015 − tcdf(X, 10^99, 13 )
4.5
E = t(0.015, 13 ) 
= (2.435845205 )  4.5  14 = 2.929531565.
14
Store this in variable E.
Compute x − E and x + E.
The 97% confidence interval is 38.07046843 <  < 43.92953157.
5. The maximum error of the estimate is to be 0.08 in a 95% confidence interval for the
proportion. What should the minimum sample size be? (Assume we cannot estimate p and q.)
E = 0.08,  = 0.05, /2 = 0.025.
For the product, pq, use the maximum value of 0.25.
n 
z(/2 ) 2  
p q
E
=
invNorm(0.975 ) 2
(0.25 ) = 150.0569854.
0.08
The minimum sample size is 151.
6. The weights of 31 loaves of bread are measured. The sample standard deviation is 0.65
ounces. Find the 98% confidence interval for the population standard deviation, showing
calculation of the critical values.
n = 31, df = n − 1 = 30, s = 0.65,  = 0.02, /2 = 0.01.
For  2L , solve 0 = 0.01 −  2 cdf(0, X, 30 ), or 0 = 0.99 −  2 cdf(X, 10^99, 30 ).
For  2R , solve 0 = 0.99 −  2 cdf(0, X, 30 ), or 0 = 0.01 −  2 cdf(X, 10^99, 30 ).
(n − 1 )s 2
 2R
< 2 <
30(0.65 ) 2
2
50.89218131 <  <
(n − 1 )s 2
 2L
0.2490559389 <  2 < 0.8476301096
The 98% confidence interval is
GO TO PAGE 3.
30(0.65 ) 2
14.95345653
Take square roots.
0.499055046 <  < 0.9206682951 .
ELEMENTARY STATISTICS
SAMPLE QUESTIONS SET TWO.
Page 3 of 3.
SHOW ALL CALCULATIONS/TI COMMANDS. GIVE ANSWERS TO AT LEAST SIX DECIMAL PLACES.
7. The weights of 39 boxes of cereal are measured. The sample standard deviation is
0.37 ounces. The sample mean is 15.9 ounces. We are concerned that the mean
weight of all boxes may be under 16 ounces.
(a) State the null and alternative hypotheses.
(b) Test the null hypothesis at the 5% level, using either p-value or classical approach,
showing calculation of the test statistic.
8. The weights of 21 loaves of bread are measured. The sample standard deviation is 0.6
ounces. We are concerned that the weights of the loaves may have a standard deviation
greater than 0.5 ounces.
(a) State the null and alternative hypotheses.
(b) Test the null hypothesis at the 10% level, using either p-value or classical approach,
showing calculation of the test statistic.
9. In the previous election, 69% of the registered voters in Midville actually voted in the election
for mayor. This year, a survey of 96 randomly chosen registered voters, found that 62 had
voted in the current election for mayor. Has there been a significant decrease in the
proportion of registered voters actually voting, at the 5% level? Show calculation of the test
statistic.
10.
Student
Carla
Rose
Kris
Dean
Alberto
Test One
78
70
82
92
81
Test Two
79
75
81
90
86
Test the hypothesis that there is no improvement in average score from test one to test two,
based on the above sample of five students, at the 1% level of significance. Show the
calculation of the test statistic.
11. One class, of 35 students, averages 77 on a test with variance, s 12 = 16. Another class, of
31 students, averages 81 on the same test, with s 22 = 25.
(a) Compute the estimated standard error of the difference of sample means.
(You have to identify exactly what this is, here.)
(b) Are the averages significantly different at the 2% level? Show calculation of the test
statistic. May use TI test for the degrees of freedom.
( 100 points, total. )
GO TO PAGE 4.