Concept explainers
Damage to grapes from bird predation is a serious problem for grape growers. The article “Experimental Method to Investigate and Monitor Bird Behavior and Damage to Vineyards” (Amer. J. of Enology and Viticulture, 2004: 288–291) reported on an experiment involving a bird-feeder table, time-lapse video, and artificial foods. Information was collected for two different bird species at both the experimental location and at a natural vineyard setting. Consider the following data on time (sec) spent on a single visit to the location.
Species | Location | n | X | SE mean |
Blackbirds | Exptl | 65 | 13.4 | 2.05 |
Blackbirds | Natural | 50 | 9.7 | 1.76 |
Silvereyes | Exptl | 34 | 49.4 | 4.78 |
Silvereyes | Natural | 46 | 38.4 | 5.06 |
a. Calculate an upper confidence bound for the true average time that blackbirds spend on a single visit at the experimental location.
b. Does it appear that true average time spent by blackbirds at the experimental location exceeds the true average time birds of this type spend at the natural location? Carry out a test of appropriate hypotheses.
c. Estimate the difference between the true average time blackbirds spend at the natural location and true average time that silvereyes spend at the natural location, and do so in a way that conveys information about reliability and precision. [Note: The sample
a.
Obtain 95% upper confidence interval for the average time that the black birds spend on a single Visit at the experimental location.
Answer to Problem 24E
The 95% confidence interval for the average time that the black birds spend on a single Visit at the experimental location is
Explanation of Solution
Given info:
The information is based on the time spent by the birds on a single visit on the experimental location.
Calculation:
Here,
The following assumptions are required for the confidence interval about mean using t- distribution to be valid.
Requirements for t-distribution to construct confidence interval about mean:
- The sample must be drawn using simple random sampling.
- The population standard deviation is unknown.
- Either the population must be approximately normal or the sample size must be greater than 30.
Thus, the requirements must be satisfied for the confidence interval about mean using t- distribution to be valid.
Critical value:
For 95% level of significance,
Degrees of freedom:
From Table A.5 of the critical values for t distribution in Appendix, the critical value
Thus, the critical value is
Confidence interval:
The confidence interval for the average time spent is,
Thus, the 95% confidence interval for the average time that the black birds spend on a single Visit at the experimental location is
b.
Check whether the average time that the black birds spend on a single Visit at the experimental location is greater than that of natural location.
Answer to Problem 24E
The conclusion is that, there is no enough evidence to infer that the average time that the black birds spend on a single Visit at the experimental location is greater than that of natural location.
Explanation of Solution
Given info:
Let
Calculation:
Here,
The test hypotheses are,
Null hypothesis:
That is, the average time that the black birds spend on a single Visit at the experimental location is same as the natural location.
Alternative hypothesis:
That is, the average time that the black birds spend on a single Visit at the experimental location is greater than that of natural location.
Assumption for the two sample t-test:
- The sample X and Y taken from the population is selected at random.
- The samples X and Y are independent of each other.
- Samples must be distributed to normal.
Here, the samples selected from the older and younger adults were selected at random and independent. Moreover, the sample size is large and distributed to normal. Hence, the assumptions are satisfied.
The degrees of freedom is,
Rounding the degrees of freedom as 120.
Test statistic for two-sample t-test:
Test statistic:
Hence, the test statistic is obtained as:
The difference between the average time spend
The test statistic value is obtained below:
Thus, the test statistic value is 1.37.
From the Appendix “Table A.5 Critical values for t Distributions”, the critical value
Decision rule:
Rejection region for a Right-tailed test:
If
Conclusion:
The level of significance is,
Here, the test statistic is less than the critical value. That is,
Thus, the decision is “fail to reject the null hypothesis”.
Thus, it can be concluded that there is no enough evidence to infer that the average time that the black birds spend on a single Visit at the experimental location is greater than that of natural location.
c.
Estimate the difference between the true average time that the black birds spend on a single Visit at the natural location and time spend by the silvereyes at natural location.
Answer to Problem 24E
The difference between the true average time that the black birds spend on a single Visit at the natural location and time spends by the silvereyes at natural location is (17.99 Sec, 39.41 Sec).
Explanation of Solution
Calculation:
Critical value:
For 95% level of significance,
From the Appendix “Table A.5 Critical values for t Distributions”, the critical value
Thus, the critical value is
Confidence interval:
The confidence interval for the average time spent is,
The 95% confidence interval for the difference between the average time that the black birds spend on a single Visit at the natural location and time spends by the silvereyes at natural location is
Interpretation:
Hence, the 95% confidence interval for the difference between the average time that the black birds spend on a single Visit at the natural location and time spend by the silvereyes at natural location is (17.99 Sec, 39.41 Sec).
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Chapter 9 Solutions
Probability and Statistics for Engineering and the Sciences
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