Q. 62 - The Practice of Statistics for AP

Question

62. Blink When two lights close together blink alternately, we “see” one light moving back and forth if the time between blinks is short. What is the
longest interval of time between blinks that preserves the illusion of motion? Ask subjects to turn a knob that slows the blinking until they “see” two lights rather than one light moving. A report gives the results in the form “mean plus or minus the standard error of the mean.” Data for $12$ subjects are summarized as $251\pm 45$ (in milliseconds).
(a) Find the sample standard deviation $s_x$ for these measurements. Show your work.
(b) Explain why the interval $251\pm 45$ is not a confidence interval.

Step-by-Step Solution

Answer

(a) The standard deviation $s_x$ for the measurements is $155.8846$

(b) The provided interval $251\pm 45$ is not a confidence interval, because $t^{*}$is unknown.

Part (a) Step 1 : Given information

Data for $12$ subjects are summarized as $251\pm 45$ (in milliseconds). To Find the sample standard deviation $s_x$ for these measurements.  

Part (a) Step 2 : Explanation

The standard deviation is divided by the square root of the sample size to get the standard error of the mean.

Standard error of mean is  $\overline{E}=\frac{s_x}{\sqrt{n}}$, and multiply with $\sqrt{n}$ on both sides.

Then, $s=\overline{E}\times \sqrt{n}$.

where, Sample size $n$ is $12$, Standard error of mean $\overline{E}=45$, and the population mean $\overline{x}$ is $251$.
$$\begin{gathered} s=\overline{E}\times \sqrt{n} \\ =45\times \sqrt{12} \\ =155.8846 \end{gathered}$$
Therefore, the standard deviation is $155.8846$.

Part (b) Step 1 : Given Information

To explain why the provided interval  $251\pm 45$ is not a confidence interval.

Part (b) step 2 : Explanation

The confidence interval is calculated by using the formula$=\overline{x}\pm \left(t^{*}\times \overline{E}\right)$.

The interval in the problem  is $\overline{x}\pm \overline{E}$.

And,  $t^{*}$ is unknown.

As a result, the provided interval is not a confidence interval.