Question

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 1

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, 100

Formula:

$\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}$  

where $x_i$ are data points, $\bar{x}$ is the mean and n is the number of observations.  

$Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}$

$Mean =\displaystyle\frac{990}{12} = 82.5$

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

$S.D = \sqrt{\frac{3539.363636}{11}} = 59.5$

We are given the following in the question:  

Population mean, μ =67

Sample mean, $\bar{x}$ = 82.5

Sample size, n = 12

Alpha, α = 0.05

Sample standard deviation, s = 59.5

First, we design the null and the alternate hypothesis

$H_{0}: \mu = 67\\H_A: \mu > 67$

We use One-tailed t test to perform this hypothesis.

b) Formula:

$t_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{s}{\sqrt{n-1}} }$

Putting all the values, we have

$t_{stat} = \displaystyle\frac{82.5 - 67}{\frac{59.5}{\sqrt{11}} } = 0.864$ Now,

$t_{critical} \text{ at 0.05 level of significance, 11 degree of freedom } = 1.795$

a) Since,                

$t_{stat} < t_{critical}$

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.