A researcher plans to use a random sample of n= 500 families to estimate the mean monthly family incomefor a large population. a 99% confidence interval based on the sample would be _________ than 90% confidence interval.
...greater than 90% confidence interval.
Probably the better answer would be "wider" than 90% confidence interval.
A 99% confidence interval would be wider than a 90% confidence interval.
In general, increasing the confidence level implies a larger interval width. This is because a higher confidence level requires more precision and therefore a larger margin of error. In this case, the researcher aims to be 99% confident in their estimation, which requires a larger range of possible values around the sample mean. Hence, the 99% confidence interval will be wider compared to the 90% confidence interval.
To determine the difference in width between a 99% confidence interval and a 90% confidence interval, we need to understand how confidence intervals are calculated and how the level of confidence affects their width.
A confidence interval is a range of values that estimates the population parameter, in this case, the mean monthly family income. The width of a confidence interval is influenced by two main factors: the sample size and the level of confidence.
A higher level of confidence requires a wider interval, as a higher level of confidence corresponds to a greater certainty that the interval contains the true population parameter. On the other hand, a smaller sample size would result in a wider interval because there is less information available to estimate the population parameter accurately.
In this scenario, the researcher plans to use a random sample of n = 500 families. Since the sample size is the same for both confidence intervals, the only factor affecting the difference in widths will be the level of confidence.
A 99% confidence interval will be wider than a 90% confidence interval. The reason is that a 99% confidence interval provides a higher level of confidence, indicating that the researcher wants to be 99% certain that the interval contains the true population mean.
To calculate the width of a confidence interval, you would typically use a formula such as:
Width = 2 * (z-value) * (standard deviation / √n)
where:
- z-value represents the critical value based on the desired level of confidence
- the standard deviation is an estimate of the population standard deviation
- n is the sample size
By keeping all other factors equal (i.e., sample size and standard deviation being constant), changing the z-value from a 90% confidence level to a 99% confidence level will result in a larger critical value. This larger critical value will cause the 99% confidence interval to be wider than the 90% confidence interval.
Therefore, the 99% confidence interval will be wider than the 90% confidence interval.