Question

The authors of a paper describe an experiment to evaluate the effect of using a cell phone on reaction time. Subjects were asked to perform a simulated driving task while talking on a cell phone. While performing this task, occasional red and green lights flashed on the computer screen. If a green light flashed, subjects were to continue driving, but if a red light flashed, subjects were to brake as quickly as possible. The reaction time (in msec) was recorded. The following summary statistics are based on a graph that appeared in the paper.

n = 47
x = 525
s = 75

(a) Construct a 95% confidence interval for μ, the mean time to react to a red light while talking on a cell phone. (Round your answers to three decimal places.)
(b) Suppose that the researchers wanted to estimate the mean reaction time to within 7 msec with 95% confidence. Using the sample standard deviation from the study described as a preliminary estimate of the standard deviation of reaction times, compute the required sample size. (Round your answer up to the nearest whole number.)

Answer 1

To construct a 95% confidence interval for the mean time to react to a red light while talking on a cell phone, use the formula CI = x ± (t * s / sqrt(n)). To calculate the required sample size with 95% confidence and a desired margin of error, use the formula n = (z * s / E)^2.

Step-by-step explanation:

(a) To construct a 95% confidence interval for the mean time to react to a red light while talking on a cell phone, we can use the formula:

CI = x ± (t * s / sqrt(n))

Where:

• CI is the confidence interval
• x is the sample mean
• t is the critical value for the desired confidence level (0.05 level of significance for a two-tailed test)
• s is the sample standard deviation
• n is the sample size

Plugging in the given values, we have:

x = 525, s = 75, n = 47

Using a t-distribution table or a t-distribution calculator, we find the critical value to be approximately 2.013.

Substituting the values into the formula, we get:

CI = 525 ± (2.013 * 75 / sqrt(47))

Simplifying this expression gives us the 95% confidence interval for the mean time to react to a red light while talking on a cell phone.

(b) To calculate the required sample size, we use the formula:

n = (z * s / E)^2

Where:

• n is the required sample size
• z is the critical value for the desired confidence level (0.05 level of significance for a two-tailed test)
• s is the sample standard deviation
• E is the desired margin of error (half of the confidence interval width)

Plugging in the given values, we have:

z = 1.96 (for 95% confidence level)

s = 75 (sample standard deviation)

E = 7 (desired margin of error)

Substituting the values into the formula, we get:

n = (1.96 * 75 / 7)^2

Rounding up to the nearest whole number, the required sample size is obtained.

Answer 2

a) The 99% confidence interval would be given by (24.409;24.979)

b) n=464

Step-by-step explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Part a

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=47-1=46`

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,46)".And we see that
`t_(\alpha/2)=2.01`

Now we have everything in order to replace into formula (1):

`525-2.01(75)/(√(47))=503.01`

`525+2.01(75)/(√(47))=546.99`

So on this case the 95% confidence interval would be given by (503.01;546.99)

Part b

The margin of error is given by this formula:

`ME=t_(\alpha/2)(s)/(√(n))` (1)

And on this case we have that ME =7 msec, we are interested in order to find the value of n, if we solve n from equation (1) we got:

`n=((t_(\alpha/2) s)/(ME))^2` (2)

The critical value for 95% of confidence interval is provided,
`t_(\alpha/2)=2.01` from part a, replacing into formula (2) we got:

`n=((2.01(75))/(7))^2 =463.79 \approx 464`

So the answer for this case would be n=464 rounded up to the nearest integer