Question

In Exercises 3.49 and 3.50 , a \(95 \%\) confidence interval is given, followed by possible values of the population parameter. Indicate which of the values are plausible values for the parameter and which are not. A \(95 \%\) confidence interval for a mean is 112.1 to \(128.2 .\) Is the value given a plausible value of \(\mu ?\) (a) \(\mu=121\) (b) \(\mu=113.4\) (c) \(\mu=105.3\)

Short answer

The plausible values are \(\mu = 121\) and \(\mu = 113.4\) as they both lie within the \(95\%\) confidence interval of 112.1 to 128.2. The value \(\mu = 105.3\) is not plausible as it falls outside of the confidence interval.

Step-by-step solution

- Assessing Plausibility for \(\mu = 121\)

Firstly, assess whether \(\mu = 121\) is a plausible value for the mean. As the value 121 lies within the given confidence interval of 112.1 to 128.2, this means it is plausible that the true population mean could be 121.

- Assessing Plausibility for \(\mu = 113.4\)

Next, assess whether \(\mu = 113.4\) is a plausible value for the mean. As the value 113.4 also falls within the confidence interval of 112.1 to 128.2, it is plausible that 113.4 could be the true mean for the population.

- Assessing Plausibility for \(\mu = 105.3\)

Lastly, assess whether \(\mu = 105.3\) is a plausible value for \(\mu\). Since value 105.3 falls outside the given confidence interval of 112.1 to 128.2, it is not plausible that the true population mean \(\mu\) is 105.3.