A Washington, D.C., "think tank" announces the typical teenager sent 67 text messages per day in 2017. To update that estimate, you phone a sample of 12 teenagers and ask them how many text messages they sent the previous day. Their responses were:51 175 47 49 44 54 145 203 21 59 42 100(a) state the decision rule for 0.05 significance level (round to 3 decimal places)(b) Compute the value of the test statistic (round to 3 decimal places)

Answer:We reject the null hypothesis and fail to accept it and update that estimate that typical teenager sent does not 67 text messages per day.       Step-by-step explanation:We are given the sample:51, 175, 47, 49, 44, 54, 145, 203, 21, 59, 42, 100Formula:[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]  where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.  [tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex][tex]Mean =\displaystyle\frac{990}{12} = 82.5[/tex]Sum of squares of differences = 992.25 + 8556.25 + 1260.25 + 1122.25 + 1482.25 + 812.25 + 3906.25 + 14520.25 + 3782.25 + 552.25 + 1640.25 + 306.25 = 3539.363636[tex]S.D = \sqrt{\frac{3539.363636}{11}} = 59.5[/tex]We are given the following in the question:  Population mean, μ =67Sample mean, [tex]\bar{x}[/tex] = 82.5Sample size, n = 12Alpha, α = 0.05Sample standard deviation, s = 59.5First, we design the null and the alternate hypothesis[tex]H_{0}: \mu = 67\\H_A: \mu > 67[/tex] We use One-tailed t test to perform this hypothesis.b) Formula:[tex]t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n-1}} }[/tex] Putting all the values, we have [tex]t_{stat} = \displaystyle\frac{82.5 - 67}{\frac{59.5}{\sqrt{11}} } = 0.864[/tex] Now, [tex]t_{critical} \text{ at 0.05 level of significance, 11 degree of freedom } = 1.795[/tex] a) Since,                [tex]t_{stat} < t_{critical}[/tex]We reject the null hypothesis and fail to accept it and update that estimate that typical teenager send more than 67 text messages per day.