9-2. Exercises

1. BASIC

Ex 1. In all exercises for this section assume that the populations are normal and have equal standard deviations.

Ex 2. Construct the confidence interval for $(μ_1−μ_2)$ for the level of confidence and the data from independent samples given.

1. 95% confidence,

   $n_1=10, \bar{x_1}=120, s_1=2 ,$

   $n_2=15, \bar{x_2}=101, s_2=4$

2. 99% confidence,

   $n_1=6, \bar{x_1}=25, s_1=1,$

   $n_2=12, \bar{x_2}=17, s_2=3$

Ex 3. Construct the confidence interval for $(μ_1−μ_2)$ for the level of confidence and the data from independent samples given.

1. 90% confidence, $n_1=28, \bar{x_1}=212, s_1=6,$ $n_2=23, \bar{x_2}=198, s_2=5$

2. 99% confidence,

   $n_1=14, \bar{x_1}=68, s_1=8,$

   $n_2=20, \bar{x_2}=43, s_2=3$

Ex 4. Construct the confidence interval for $(μ_1−μ_2)$ for the level of confidence and the data from independent samples given.

1. 99.9% confidence,

   $n_1=35, \bar{x_1}=6.5, s_1=0.2 ,$

   $n_2=20, \bar{x_2}=6.2, s_2=0.1$

2. 99% confidence,

   $n_1=18, \bar{x_1}=77.3, s_1=1.2 ,$

   $n_2=32, \bar{x_2}=75.0, s_2=1.6$

Ex 5. Construct the confidence interval for $(μ_1−μ_2)$ for the level of confidence and the data from independent samples given.

1. 99.5% confidence,

   $n_1=40, \bar{x_1}=85.6, s_1=2.8,$

   $n_2=20, \bar{x_2}=73.1, s_2=2.1$

2. 99.9% confidence,

   $n_1=25, \bar{x_1}=215, s_1=7,$

   $n_2=35, \bar{x_2}=185, s_2=12$

Ex 6. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

1. Test $H_0:μ1−μ2=11 \space vs. \space H_a:μ1−μ2>11, @ α=0.025,$

   $n_1=6, \bar{x_1}=32, s_1=2 ,$

   $n_2=11, \bar{x_2}=19, s_2=1$

2. Test $H_0:μ1−μ2=26 \space vs. \space H_a:μ1−μ2 \ne 26, @ α=0.05,$

   $n_1=17, \bar{x_1}=166, s_1=4,$

   $n_2=24, \bar{x_2}=138, s_2=3$

Ex 7. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

1. Test $H_0:μ1−μ2=40 \space vs. \space H_a:μ1−μ2<40, @ α=0.10,$

   $n_1=14, \bar{x_1}=289, s_1=11,$

   $n_2=12, \bar{x_2}=254, s_2=9$

2. Test $H_0:μ1−μ2=21 \space vs. \space H_a:μ1−μ2\ne 21, @ α=0.05,$

   $n_1=23, \bar{x_1}=130, s_1=6,$

   $n_2=27, \bar{x_2}=113, s_2=8$

Ex 8. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

1. Test $H_0:μ1−μ2=-15 \space vs. \space H_a:μ1−μ2<-15, @ α=0.10,$

   $n_1=30, \bar{x_1}=42, s_1=7,$

   $n_2=12, \bar{x_2}=60, s_2=5$

2. Test $H_0:μ1−μ2=103 \space vs. \space H_a:μ1−μ2 \ne 103, @ α=0.10,$

   $n_1=17, \bar{x_1}=711, s_1=28,$

   $n_2=32, \bar{x_2}=598, s_2=21$

Ex 9. Perform the test of hypotheses indicated, using the data from independent samples given. Use the critical value approach.

1. Test $H_0:μ1−μ2=75 \space vs. \space H_a:μ1−μ2>75, @ α=0.025,$

   $n_1=45, \bar{x_1}=674, s_1=18,$

   $n_2=29, \bar{x_2}=591, s_2=13$

2. Test $H_0:μ1−μ2=20 \space vs. \space H_a:μ1−μ2 \ne 20, @ α=0.005,$

   $n_1=30, \bar{x_1}=137, s_1=8,$

   $n_2=19, \bar{x_2}=166, s_2=11$

Ex 10. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach. (The p-value can be only approximated.)

1. Test $H_0:μ1−μ2=12 \space vs. \space H_a:μ1−μ2>12, @ α=0.01,$

   $n_1=20, \bar{x_1}=133, s_1=7,$

   $n_2=10, \bar{x_2}=115, s_2=5$

2. Test $H_0:μ1−μ2=46 \space vs. \space H_a:μ1−μ2\ne 46, @ α=0.10,$

   $n_1=24, \bar{x_1}=586, s_1=11,$

   $n_2=27, \bar{x_2}=535, s_2=13$

Ex 11. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach. (The p-value can be only approximated.)

1. Test $H_0:μ1−μ2=38 \space vs. \space H_a:μ1−μ2<38, @ α=0.01,$

   $n_1=12, \bar{x_1}=464, s_1=5,$

   $n_2=10, \bar{x_2}=432, s_2=6$

2. Test $H_0:μ1−μ2=4 \space vs. \space H_a:μ1−μ2 \ne 4, @ α=0.005,$

   $n_1=14, \bar{x_1}=68, s_1=2,$

   $n_2=17, \bar{x_2}=67, s_2=3$

Ex 12. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach. (The p-value can be only approximated.)

1. Test $H_0:μ1−μ2=50 \space vs. \space H_a:μ1−μ2>50, @ α=0.01,$

   $n_1=30, \bar{x_1}=2681, s_1=8,$

   $n_2=27, \bar{x_2}=625, s_2=8$

2. Test $H_0:μ1−μ2=35 \space vs. \space H_a:μ1−μ2 \ne35, @ α=0.10,$

   $n_1=36, \bar{x_1}=325, s_1=11,$

   $n_2=29, \bar{x_2}=286, s_2=7$

Ex 13. Perform the test of hypotheses indicated, using the data from independent samples given. Use the p-value approach. (The p-value can be only approximated.)

1. Test $H_0:μ1−μ2=-4 \space vs. \space H_a:μ1−μ2< =4, @ α=0.05,$

   $n_1=40, \bar{x_1}=80, s_1=5,$

   $n_2=25, \bar{x_2}=87, s_2=5$

2. Test $H_0:μ1−μ2=21 \space vs. \space H_a:μ1−μ2\ne 21, @ α=0.01,$

   $n_1=15, \bar{x_1}=192, s_1=12 ,$

   $n_2=34, \bar{x_2}=180, s_2=8$

2. APPLICATIONS

Ex 14. A county environmental agency suspects that the fish in a particular polluted lake have elevated mercury level. To confirm that suspicion, five striped bass in that lake were caught and their tissues were tested for mercury. For the purpose of comparison, four striped bass in an unpolluted lake were also caught and tested. The fish tissue mercury levels in mg/kg are given below. Sample 1  Sample 2

(from polluted lake) (from unpolluted lake)

0.580 0.382 0.711 0.276

0.571 0.570

0.666 0.366 

0.598

1. Construct the 95% confidence interval for the difference in the population means based on these data.

2. Test, at the 5% level of significance, whether the data provide sufficient evidence to conclude that fish in the polluted lake have elevated levels of mercury in their tissue.

Ex 15. A genetic engineering company claims that it has developed a genetically modified tomato plant that yields on average more tomatoes than other varieties. A farmer wants to test the claim on a small scale before committing to a full-scale planting. Ten genetically modified tomato plants are grown from seeds along with ten other tomato plants. At the season’s end, the resulting yields in pound are recorded as below. Sample 1 Sample 2 (genetically modified) (regular) 20 21 23 21 27 22 25 18 25 20 25 20 27 18 23 25 24 23 22 20

1. Construct the 99% confidence interval for the difference in the population means based on these data.

2. Test, at the 1% level of significance, whether the data provide sufficient evidence to conclude that the mean yield of the genetically modified variety is greater than that for the standard variety.

Ex 16. The coaching staff of a professional football team believes that the rushing offense has become increasingly potent in recent years. To investigate this belief, 20 randomly selected games from one year’s schedule were compared to 11 randomly selected games from the schedule five years later. The sample information on rushing yards per game (rypg) is summarized below.

	n	$\bar{x}$	s
rypg previously	20	112	24
rypg recently	11	114	21

1. Construct the 95% confidence interval for the difference in the population means based on these data.

2. Test, at the 5% level of significance, whether the data on rushing yards per game provide sufficient evidence to conclude that the rushing offense has become more potent in recent years.

Ex 17. The coaching staff of professional football team believes that the rushing offense has become increasingly potent in recent years. To investigate this belief, 20 randomly selected games from one year’s schedule were compared to 11 randomly selected games from the schedule five years later. The sample information on passing yards per game (pypg) is summarized below.

	n	$\bar{x}$	s
pypg previously	20	203	38
pypg recently	11	232	33

1. Construct the 95% confidence interval for the difference in the population means based on these data.

2. Test, at the 5% level of significance, whether the data on passing yards per game provide sufficient evidence to conclude that the passing offense has become more potent in recent years.

Ex 18. A university administrator wishes to know if there is a difference in average starting salary for graduates with master’s degrees in engineering and those with master’s degrees in business. Fifteen recent graduates with master’s degree in engineering and 11 with master’s degrees in business are surveyed and the results are summarized below.

	n	$\bar{x}$	s
Engineering	15	68,535	1627
Business	11	63,230	2033

1. Construct the 90% confidence interval for the difference in the population means based on these data.

2. Test, at the 10% level of significance, whether the data provide sufficient evidence to conclude that the average starting salaries are different.

Ex 19. A gardener sets up a flower stand in a busy business district and sells bouquets of assorted fresh flowers on weekdays. To find a more profitable pricing, she sells bouquets for 15 dollars each for ten days, then for 10 dollars each for five days. Her average daily profit for the two different prices are given below.

	n	$\bar{x}$	s
$15	10	171	26
$10	5	198	29

1. Construct the 90% confidence interval for the difference in the population means based on these data.

2. Test, at the 10% level of significance, whether the data provide sufficient evidence to conclude the gardener’s average daily profit will be higher if the bouquets are sold at $10 each.