Question

The following data represent soil water content (percentage of water by volume) for independent random samples of soil taken from two experimental fields growing bell peppers. Soil water content from field I: x1; n1 = 72 15.1 11.3 10.1 10.8 16.6 8.3 9.1 12.3 9.1 14.3 10.7 16.1 10.2 15.2 8.9 9.5 9.6 11.3 14.0 11.3 15.6 11.2 13.8 9.0 8.4 8.2 12.0 13.9 11.6 16.0 9.6 11.4 8.4 8.0 14.1 10.9 13.2 13.8 14.6 10.2 11.5 13.1 14.7 12.5 10.2 11.8 11.0 12.7 10.3 10.8 11.0 12.6 10.8 9.6 11.5 10.6 11.7 10.1 9.7 9.7 11.2 9.8 10.3 11.9 9.7 11.3 10.4 12.0 11.0 10.7 8.9 11.2 Soil water content from field II: x2; n2 = 80 12.2 10.2 13.6 8.1 13.5 7.8 11.8 7.7 8.1 9.2 14.1 8.9 13.9 7.5 12.6 7.3 14.9 12.2 7.6 8.9 13.9 8.4 13.4 7.1 12.4 7.6 9.9 26.0 7.3 7.4 14.3 8.4 13.2 7.3 11.3 7.5 9.7 12.3 6.9 7.6 13.8 7.5 13.3 8.0 11.3 6.8 7.4 11.7 11.8 7.7 12.6 7.7 13.2 13.9 10.4 12.9 7.6 10.7 10.7 10.9 12.5 11.3 10.7 13.2 8.9 12.9 7.7 9.7 9.7 11.4 11.9 13.4 9.2 13.4 8.8 11.9 7.1 8.5 14.0 14.5 (a) Use a calculator with mean and standard deviation keys to calculate x1, s1, x2, and s2. (Round your answers to two decimal places.) (b) Let ?1 be the population mean for x1 and let ?2 be the population mean for x2. Find a 95% confidence interval for ?1 ? ?2. (Round your answers to two decimal places.) lower limit upper limit (c) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? All negative? Of different signs? At the 95% level of confidence, is the population mean soil water content of the first field higher than that of the second field? Because the interval contains only positive numbers, we can say that the mean soil water content of the first field is higher. Because the interval contains both positive and negative numbers, we cannot say that the mean soil water content of the first field is higher. We can not make any conclusions using this confidence interval. Because the interval contains only negative numbers, we can say that the mean soil water content of the second field is higher. (d) Which distribution did you use (standard normal or Student's t)? Why? The Student's t-distribution was used because ?1 and ?2 are known. The standard normal distribution was used because ?1 and ?2 are unknown. The Student's t-distribution was used because ?1 and ?2 are unknown. The standard normal distribution was used because ?1 and ?2 are known. Do you need information about the soil water content distributions? Both samples are small, so information about the distributions is not needed. Both samples are small, so information about the distributions is needed. Both samples are large, so information about the distributions is not needed. Both samples are large, so information about the distributions is needed. (e) Use ? = 0.01 to test the claim that the population mean soil water content of the first field is higher than that of the second. (i) What is the level of significance? State the null and alternate hypotheses. H0: ?1 = ?2; H1: ?1 < ?2 H0: ?1 = ?2; H1: ?1 ? ?2 H0: ?1 ? ?2; H1: ?1 = ?2 H0: ?1 = ?2; H1: ?1 > ?2 (ii) What sampling distribution will you use? What assumptions are you making? The standard normal. We assume that both population distributions are approximately normal with known standard deviations. The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations. The Student's t. We assume that both population distributions are approximately normal with known standard deviations. The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations. What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate. (Test the difference ?1 ? ?2. Do not use rounded values. Round your answer to three decimal places.) (iii) Find (or estimate) the P-value. P-value > 0.250 0.125 < P-value < 0.250 0.050 < P-value < 0.125 0.025 < P-value < 0.050 0.005 < P-value < 0.025 P-value < 0.005 Sketch the sampling distribution and show the area corresponding to the P-value.

Answer 1

mean of x1 = 12.24658

mean of x2 = 11.51358

standard deviation of x1 = s1 = 7.385742

standard deviation of x2 = s2 = 8.273493

c) 95% confidence interval for ?1 ? ?2 =>

( -1.760078 3.226068 )

there is 95% chance the values lies between these interval.

the interval is of different sign. lower value= -1.760078 , upper value = 3.226068

At the 95% level of confidence, is the population mean soil water content of the first field higher than that of the second field Because the interval contains both positive and negative numbers, we cannot say that the mean soil water content of the first field is higher. We can not make any conclusions using this confidence interval.

d) The Student's t-distribution was used here. because ?1 and ?2 are unknown.

e) we use ? = 0.01 level of significance.

to test the claim that the population mean soil water content of the first field is higher than that of the second.

i) null hypothesis = H0: ?1 = ?2; alternative hypothesis = H1: ?1 > ?2

we use t test for the distribution

test statistics = t = 0.58088

p value =  0.2811

Decision rule :-

i) if p value > 0.01 l.o.s then accept null hypothesis H0 at level of significance 0.01

ii) if p value < 0.01 l.o.s if then reject  null hypothesis H0 at level of significance 0.01

Result :-  p value =  0.2811 > 0.01 level of significance. so accept null hypothesis H0 at level of significance 0.01

conclusion :- the population mean soil water content of the first field is not higher than that of the second.