Respuesta :
The confidence interval is [tex]\fbox{(5.8, 8.1)}[/tex] and [tex]\fbox{\text{Option A}}[/tex] is correct.
Further Explanation:
Given:
The least value is [tex]1[/tex] that is the least attractive.
The highest value is [tex]10[/tex] that is the most attractive.
The observations are,
7, 7, 3, 8, 5, 6, 6, 9, 9, 8, 6, 9.
Calculation:
The sum of all observations is [tex]83[/tex].
The population mean is [tex]\mu[/tex].
The standard deviation [tex]s[/tex] is [tex]1.832[/tex].
The sample mean [tex]\bar^{X}[/tex] is [tex]6.92[/tex].
Level of significance [tex]\alpha[/tex] = [tex]5\%[/tex].
Formula for confidence interval = [tex]\left( \bar{X} \pm t_{n-1, \frac{\alpha}{2}\%} \frac{s}{\sqrt{n}} \right)[/tex]
Confidence interval =[tex]\left( 6.92 \pm t_{12-1, \frac{5}{2}\%} \frac{1.832}{\sqrt{12}} \right)[/tex]
The value of [tex]t_{11, \frac{5}{2}\%[/tex]=[tex]2.201[/tex]
Confidence interval = [tex]( 6.92 \pm 2.201}\times \frac{1.832}{\sqrt{12}}) \right)[/tex]
Confidence interval = [tex]( 6.92 - 2.201}\times 0.5288 ,6.92 + 2.201}\times \0.5288) \right)[/tex]
Confidence interval = [tex](6.92-1.1639,6.92+1.1639)[/tex]
Confidence interval = [tex]\fbox{(5.8, 8.1)}[/tex]
The [tex]95\%[/tex] confidence interval gives us an idea that [tex]95\%[/tex] chances of the true mean or population mean lies in the interval.
A. We are [tex]95\%[/tex] confident that the interval from [tex]5.8[/tex] to [tex]8.1[/tex] actually contains the true mean of attractiveness rating of all adult females.
B. The results tell nothing about the population of all adult females, because participants in speed dating are not a representative sample of the population of all adult females.
C. We are confident that [tex]95\%[/tex] of all adult females have attractiveness ratings between [tex]5.8[/tex] and [tex]8.1[/tex].
[tex]\fbox{\text{Option A}}[/tex] is Correct as we are [tex]95\%[/tex] confident that the interval from [tex]5.8[/tex] to [tex]8.1[/tex] actually contains the true mean of attractiveness rating of all adult females.
Option B is not correct as the confidence interval tells us about the population mean.
Option C is not correct as the individual rating can be more than [tex]5.8[/tex] or less than the [tex]8.1[/tex].
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Answer Details:
Grade: College Statistics
Subject: Mathematics
Chapter: Confidence Interval
Keywords:
Probability, Statistics, Speed dating, Females rating, Confidence interval, t-test, Level of significance , Normal distribution, Central Limit Theorem, t-table, Population mean, Sample mean, Standard deviation, Symmetric, Variance.
The confidence interval is [tex](5.8\;,\;8.1)[/tex]
The least value is [tex]1[/tex] that is the least attractive.
The highest value is [tex]10[/tex] that is the most attractive.
The observations are [tex](7, 7, 3, 8, 5, 6, 6, 9, 9, 8, 6, 9)[/tex].
The sum of all observations is [tex]83[/tex], population mean is [tex]\mu[/tex], standard deviation [tex]\delta[/tex] is [tex]1.832[/tex] , sample mean [tex]\bar{X}[/tex] is [tex]6.92[/tex] and Level of significance [tex]\alpha =5[/tex] %
Formula for confidence interval is:
[tex]=\bar{X}\pm t_{n-1,\frac{\alpha }{2}}\frac{\delta }{^{\sqrt{n}}}\;={6.92}\pm t_{12-1,\frac{5}{2}}\frac{1.832}{^{\sqrt{12}}}[/tex] ([tex]\because[/tex]The value of [tex]t_{n-1,\frac{\alpha }{2}}[/tex] is [tex]2.201[/tex])
[tex]\Rightarrow CI=(6.92\pm 2.201\times \frac{1.832}{\sqrt{12}})[/tex]
[tex]=(6.92-2.201\times 0.5288,6.92+2.201\times 0.5288)\\ \\ \Rightarrow CI=(5.8,8.1)[/tex]
Learn more about confidence interval.
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