Get expert advice and community support for your questions on IDNLearn.com. Discover detailed answers to your questions with our extensive database of expert knowledge.
Sagot :
To determine the correct interpretation of the given [tex]$95 \%$[/tex] confidence interval for the mean difference in height of pairs of identical twins, we need to understand what a confidence interval represents.
A [tex]$95 \%$[/tex] confidence interval provides a range of values within which we can be [tex]$95 \%$[/tex] confident that the true population parameter lies. In this case, the parameter of interest is the true mean difference in height between Twin 1 and Twin 2.
Given the data, we need to interpret the confidence interval [tex]\((-0.823, 0.573)\)[/tex]. Here's a step-by-step solution:
1. Identify the Population Parameter: We are interested in the mean difference in height between pairs of identical twins. Note that this is a population parameter.
2. Sample and Confidence Interval: The sample consists of heights from 8 pairs of twins. The calculated [tex]$95 \%$[/tex] confidence interval for the mean difference (Twin 1 - Twin 2) is [tex]\((-0.823, 0.573)\)[/tex].
3. Interpretation of the Confidence Interval:
- The confidence interval captures the range within which the true mean difference in height between Twin 1 and Twin 2 is likely to fall.
- Specifically, with [tex]$95 \%$[/tex] confidence, we believe the true mean difference is between [tex]\(-0.823\)[/tex] inches and [tex]\(0.573\)[/tex] inches.
4. Finding the Correct Choice:
- The interval needs to capture the true mean difference, not just the mean heights.
- The interval should pertain to the population parameter, not the sample statistics.
Reviewing the options:
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins." - This option is incorrect because it speaks about the height of twins, not the difference in their heights.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins in this sample." - This option is incorrect because it incorrectly describes the interval as capturing the sample mean, not the population parameter.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins." - This option is correct because it correctly states that the interval captures the true mean difference in height.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of the twins in this sample." - This option is incorrect because it refers to the sample, not the population.
Therefore, the correct interpretation of the [tex]$95 \%$[/tex] confidence interval is: "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins."
A [tex]$95 \%$[/tex] confidence interval provides a range of values within which we can be [tex]$95 \%$[/tex] confident that the true population parameter lies. In this case, the parameter of interest is the true mean difference in height between Twin 1 and Twin 2.
Given the data, we need to interpret the confidence interval [tex]\((-0.823, 0.573)\)[/tex]. Here's a step-by-step solution:
1. Identify the Population Parameter: We are interested in the mean difference in height between pairs of identical twins. Note that this is a population parameter.
2. Sample and Confidence Interval: The sample consists of heights from 8 pairs of twins. The calculated [tex]$95 \%$[/tex] confidence interval for the mean difference (Twin 1 - Twin 2) is [tex]\((-0.823, 0.573)\)[/tex].
3. Interpretation of the Confidence Interval:
- The confidence interval captures the range within which the true mean difference in height between Twin 1 and Twin 2 is likely to fall.
- Specifically, with [tex]$95 \%$[/tex] confidence, we believe the true mean difference is between [tex]\(-0.823\)[/tex] inches and [tex]\(0.573\)[/tex] inches.
4. Finding the Correct Choice:
- The interval needs to capture the true mean difference, not just the mean heights.
- The interval should pertain to the population parameter, not the sample statistics.
Reviewing the options:
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins." - This option is incorrect because it speaks about the height of twins, not the difference in their heights.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean height of twins in this sample." - This option is incorrect because it incorrectly describes the interval as capturing the sample mean, not the population parameter.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins." - This option is correct because it correctly states that the interval captures the true mean difference in height.
- "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of the twins in this sample." - This option is incorrect because it refers to the sample, not the population.
Therefore, the correct interpretation of the [tex]$95 \%$[/tex] confidence interval is: "The doctor can be [tex]$95 \%$[/tex] confident that the interval from -0.823 inches to 0.573 inches captures the true mean difference in the height of twins."
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.
