Question

ch 21 is there a difference in the amount of airborne bacteria between carpeted and uncarpeted rooms? in an experiment, seven rooms were carpeted and seven were left uncarpeted. the rooms are similar in size and function. after a suitable period, the concentration of bacteria in the air was measured (in units of bacteria per cubic foot) in all of these rooms. the data and summaries are provided: xbar s carpeted rooms: 184 22.0 uncarpeted rooms: 175 16.9 the researcher wants to investigate whether the presence of carpet is associated with either an increase or decrease in the mean bacterial concentration in air. the numerical value of the two - sample t statistic for this test is 3.818. 0.414. 0.858. 1.312.

Explanation:

Step1: Recall the formula for two - sample t statistic

The formula for the two - sample t statistic is \(t=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{s_1^{2}}{n_1}+\frac{s_2^{2}}{n_2}}}\)

Step2: Identify the values from the problem

We have \(\bar{x}_1 = 184\), \(s_1=22.0\), \(n_1 = 7\), \(\bar{x}_2=175\), \(s_2 = 16.9\), \(n_2=7\)

Step3: Calculate the numerator

\(\bar{x}_1-\bar{x}_2=184 - 175=9\)

Step4: Calculate the denominator

First, calculate \(\frac{s_1^{2}}{n_1}=\frac{22^{2}}{7}=\frac{484}{7}\approx69.14\) and \(\frac{s_2^{2}}{n_2}=\frac{16.9^{2}}{7}=\frac{285.61}{7}\approx40.80\)

Then \(\sqrt{\frac{s_1^{2}}{n_1}+\frac{s_2^{2}}{n_2}}=\sqrt{69.14 + 40.80}=\sqrt{109.94}\approx10.48\)

Step5: Calculate the t statistic

\(t=\frac{9}{10.48}\approx0.858\)

Answer:

0.858