Question

Braking Reaction Times: Confidence Intervals

a. Construct a 99% confidence interval estimate of the mean braking reaction time of males, construct a 99% confidence interval estimate of the mean braking reaction time of females, then compare the results.

b. Construct a 99% confidence interval estimate of the difference between the mean braking reaction time of males and the mean braking reaction time of females.

c. Which is better for comparing the mean reaction times of males and females: the results from part (a) or the results from part (b)?

Short answer

a. The 99% confidence interval for the mean braking reaction time for males is between 40.1 and 48.7. The 99% confidence interval for the mean braking reaction time for females is between 47.2 and 61.4. The mean braking time of females is not different from that of males.

b. The 99% confidence interval for the mean difference between the braking reaction time for males and females is between –1.8 and –18.0.

c. The results in part (b) are better than in part (a).

Step-by-step solution

Given information

Refer to Exercise 6 for the data on the braking reaction times of males and females.

Males	28	30	31	34	34	36	36	36	36
	38	39	40	40	40	40	41	41	41
	42	42	44	46	47	48	48	49	51
	53	54	54	56	57	60	61	61	63
Females	22	24	34	36	36	37	39	41	41
	43	43	45	45	47	53	54	54	55
	56	57	57	57	58	61	62	63	66
	67	68	71	72	76	77	78	79	80

The number of males and females sampled:

\(\begin{aligned} {n_M} &= {n_F}\\ &= 36\end{aligned}\)

Determine the sample measures for males and females

Sample mean time for males:

\(\begin{aligned} {{\bar x}_M} &= \frac{{\sum {{x_i}} }}{{{n_m}}}\\ &= \frac{{28 + 30 + 31 + ... + 63}}{{36}}\\ &= 44.3611\end{aligned}\)

Sample mean time for females:

\(\begin{aligned}{\bar x}_F &= \frac{{\sum {{x_j}} }}{{{n_F}}}\\ &= \frac{{22 + 24 + 34 + ... + 80}}{{36}}\\ &= 54.2778\end{aligned}\)

The sample standard deviation for males is computed below:

\(\begin{aligned} {s_M} &= \sqrt {\frac{{\sum {{{\left( {{x_i} - {{\bar x}_M}} \right)}^2}} }}{{{n_M} - 1}}} \\ &= \sqrt {\frac{{{{\left( {28 - 44.36} \right)}^2} + {{\left( {30 - 44.36} \right)}^2} + ... + {{\left( {63 - 44.36} \right)}^2}}}{{36 - 1}}} \\ &= 9.4722\end{aligned}\)

The sample standard deviation for females is computed below:

\(\begin{aligned} {s_F} &= \sqrt {\frac{{\sum {{{\left( {{x_j} - {{\bar x}_F}} \right)}^2}} }}{{{n_F} - 1}}} \\ &= \sqrt {\frac{{{{\left( {22 - 54.28} \right)}^2} + {{\left( {24 - 54.28} \right)}^2} + ... + {{\left( {80 - 54.28} \right)}^2}}}{{36 - 1}}} \\ &= 15.6106\end{aligned}\)

Determine the one sample confidence interval

a.

Confidence interval for the mean reaction time of males:

The 99% confidence interval implies that the significance level is 0.01.

The degree of freedom is computedbelow:

\(\begin{aligned} df &= {n_M} - 1\\ &= 36 - 1\\ &= 35\end{aligned}\)

Using the t-distribution table at 35 degrees of freedom and 0.01significance level, you get the two-tailed critical value as\({t_{\frac{\alpha }{2}}} = 2.7238\).

The margin of error is computed below:

\(\begin{aligned} E &= {t_{\frac{\alpha }{2}}} \times \frac{{{s_M}}}{{\sqrt {{n_M}} }}\\ &= 2.7238 \times \frac{{9.4722}}{{\sqrt {36} }}\\ &= 4.3001\end{aligned}\)

The 99% confidence interval is computed below:

\(\begin{aligned}{l}{{\bar x}_M} - E < {\mu _M} < {{\bar x}_M} + E &= 44.3611 - 4.3001 < {\mu _M} < 44.3611 + 4.3001\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; &= 40.0610 < {\mu _M} < 48.6612\end{aligned}\)

Thus, the 99% confidence interval for the mean braking reaction time for males is between 40.1 and 48.7.

Confidence interval for the mean reaction time of females:

The degree of freedom is computed below:

\(\begin{aligned}{c}df &= {n_F} - 1\\ &= 36 - 1\\ &= 35\end{aligned}\)

Using the t-distribution table at 35 degrees of freedom, you get the critical value as\({t_{\frac{\alpha }{2}}} = 2.7238\).

The margin of error is computed below:

\(\begin{aligned} E &= {t_{\frac{\alpha }{2}}} \times \frac{{{s_F}}}{{\sqrt {{n_F}} }}\\ &= 2.7238 \times \frac{{15.6106}}{{\sqrt {36} }}\\ &= 7.0867\end{aligned}\)

The 99% confidence interval is computed below:

\(\begin{aligned}{l}{{\bar x}_F} - E < {\mu _F} < {{\bar x}_F} + E &= 54.2778 - 7.0867 < {\mu _F} < 54.2778 + 7.0867\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; &= 47.1911 < {\mu _F} < 61.3645\end{aligned}\)

Thus, the 99% confidence interval for the mean braking reaction time for females is between 47.2 and 61.4.

Since the two intervals overlap, it can be concluded that the mean braking time of females is not significantly different from that of males.

Determine the confidence interval for two sample means

b.

The degree of freedom is computedbelow:

\(\begin{aligned}{c}df &= \frac{{{{\left( {\frac{{s_M^2}}{{{n_M}}} + \frac{{s_F^2}}{{{n_F}}}} \right)}^2}}}{{\frac{{{{\left( {\frac{{s_M^2}}{{{n_M}}}} \right)}^2}}}{{{n_M} - 1}} + \frac{{{{\left( {\frac{{s_F^2}}{{{n_F}}}} \right)}^2}}}{{{n_F} - 1}}}}\\ &= \frac{{{{\left( {\frac{{{{9.4722}^2}}}{{36}} + \frac{{{{15.6106}^2}}}{{36}}} \right)}^2}}}{{\frac{{{{\left( {\frac{{{{9.4722}^2}}}{{36}}} \right)}^2}}}{{36 - 1}} + \frac{{{{\left( {\frac{{{{15.6106}^2}}}{{36}}} \right)}^2}}}{{36 - 1}}}}\\ &= 58\end{aligned}\)

Thus, the degree of freedom is 58.

Using the t-distribution table at 58 degrees of freedom, you get the critical value as\({t_{\frac{\alpha }{2}}} = 2.6633\) .

The margin of error is computed below:

\(\begin{aligned} E &= {t_{\frac{\alpha }{2}}} \times \sqrt {\frac{{s_M^2}}{{{n_M}}} + \frac{{s_F^2}}{{{n_F}}}} \\ &= 2.6633 \times \sqrt {\frac{{{{9.4722}^2}}}{{36}} + \frac{{{{15.6106}^2}}}{{36}}} \\ &= 8.1051\end{aligned}\)

The 99% confidence interval for the mean difference is computedbelow:

\(\begin{aligned} C.I. &= \left( {{{\bar x}_M} - {{\bar x}_F}} \right) - E < {\mu _M} - {\mu _F} < \left( {{{\bar x}_M} - {{\bar x}_F}} \right) + E\\ &= \left( {44.3611 - 54.2778} \right) - 8.1051 < {\mu _M} - {\mu _F} < \left( {44.3611 - 54.2778} \right) + 8.1051\\\; &= - 18.0218 < {\mu _M} - {\mu _F} < - 1.8116\\ &\approx - 18.02 < {\mu _M} - {\mu _F} < - 1.8\end{aligned}\)

Thus, the 99% confidence interval for the mean difference between braking reaction time for males and females is between –18.0 and –1.8.

The difference of the two means implies that the mean reaction time of males is less than that of females. Since it does not include zero, it also indicates that there is a significant difference between the two reaction times of the two groups.

Compare the results in part (a) and part (b)

The result in part (b) isbetter as it helps to compare the reaction times ofmales and females.

On the other hand, the result in part (a) only estimates the mean range in individual groups.