A car dealer wants to get information about the number of years car owners keep their cars. A random sample of 25 car owners resulted in ?̅ = 7.01 years, and ? = 3.74 years. Assume thatthe sample is drawn from a normally distributed population.
All other information remaining unchanged, which of the following would produce a wider interval than the 95% confidence interval constructed?
(a) The sample size is 29 instead of 25
(b) The sample size is 10 instead of 25
(c) Compute a 80% confidence interval rather than a 95% confidence interval.
(d) Compute a 90% confidence interval rather than a 95% confidence interval. (e) The sample standard deviation is computed to be 1.52 instead of 3.74.Solution:
Identifying which will produce a wider interval than the 95% confidence interval constructed.
Sample size and confidence interval:
The sample size indicates the amount of information we have and therefore, it determines the precision or the level of confidence that one is having in the sample estimates. Estimates are associated with some level of uncertainty. As the sample size increases, the level of uncertainty reduces providing more precise estimates. Thus, as the sample size increases, the confidence in the estimate obtained increases, uncertainty decreases and we will have more precision.
As the sample size increases, the width of the confidence interval decreases. That is, smaller sample sizes will generate wider intervals. There is an inverse square root relationship between confidence intervals and sample sizes.
Thus, a small sample size produces a wider interval. Here, a sample size of 10 will produce wider intervals than intervals of sample sizes 25 and 29.
Confidence level and confidence interval:
The confidence interval represents the level of uncertainty associated with an estimate. It is the chance that the confidence interval (margin of error) will contain the true value of the parameter that one is trying to estimate. A higher confidence level leads to a wider confidence interval with less precise estimates (that is, greater margin of error). The 95% confidence interval is wider than the 90% confidence interval and an 80% confidence interval.
Thus, a 95% confidence interval will be wider than 80% and 90% confidence intervals.
Sample standard deviation and confidence interval:
The standard error is the standard deviation divided by the square root of the sample size. As the sample size increases, the standard error decreases and thus the width of the confidence interval reduces. To halve the standard error, one must quadruple...
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