Answer:
a) 0.2667
b) (0.2167,0.3167)
c) We cannot conclude that its market share is more than 25%.
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 300
Number of passengers who used airlines, x = 80
a) point estimate of the proportion of the market that uses this particular airline.
[tex]\hat{p} = \dfrac{x}{n} = \dfrac{80}{300} = $0.2667[/tex]
b) 95% confidence interval
[tex]\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
[tex]z_{critical}\text{ at}~\alpha_{0.05} = \pm 1.96[/tex]
Putting the values, we get:
[tex]0.2667 \pm 1.96(\sqrt{\frac{0.2667(1-0.2667)}{300}}) = 0.2667 \pm 0.0500\\\\=(0.2167,0.3167)[/tex]
c) First, we design the null and the alternate hypothesis
[tex]H_{0}: p = 0.25\\H_A: p > 0.25[/tex]
Formula:
[tex]\hat{p} = \dfrac{x}{n} = \dfrac{52}{400} = 0.13[/tex]
[tex]z = \dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
Putting the values, we get,
[tex]z = \displaystyle\frac{0.2667-0.25}{\sqrt{\frac{0.25(1-0.25)}{300}}} = 0.6664[/tex]
Now, we calculate the p-value from the table.
P-value = 0.252
Since the p-value is greater than the significance level, we fail to reject the null hypothesis and accept it.
Thus, we cannot conclude that its market share is more than 25%.
