Respuesta :
Answer:
a) The 95% confidence interval for the true proportion of Canadian chefs who believe slow cooking is the most popular preparation method is (0.5457, 0.6485).
b) The 95% confidence interval for the true proportion of Canadian chefs who believe gluten-free is the most popular culinary theme us (0.2574, 0.3540).
c) The 95% confidence interval for the true proportion of Canadian chefs who believe bite-size desserts are the most popular dessert item is (0.233, 0.327).
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of \pi, and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
a. Find a 95% confidence interval for the true proportion of Canadian chefs who believe slow cooking is the most popular preparation method.
There are 350 professionals, so [tex]n = 350[/tex].
209 chefs selected slow cooking as the most popular preparation method, so [tex]\pi = \frac{209}{350} = 0.5971[/tex].
We have that [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{350}} = 0.5971 - 1.96\sqrt{\frac{0.5971*0.4029}{350}} = 0.5457[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{350}} = 0.5971 + 1.96\sqrt{\frac{0.5971*0.4029}{350}} = 0.6485[/tex]
The 95% confidence interval for the true proportion of Canadian chefs who believe slow cooking is the most popular preparation method is (0.5457, 0.6485).
b. Find a 95% confidence interval for the true proportion of Canadian chefs who believe gluten-free is the most popular culinary theme.
107 chefs selected gluten-free as the most popular culinary theme, so [tex]\pi = \frac{107}{350} = 0.3057[/tex]
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{350}} = 0.3057 - 1.96\sqrt{\frac{0.3057*0.6943}{350}} = 0.2574[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{350}} = 0.3057 + 1.96\sqrt{\frac{0.3057*0.6943}{350}} = 0.3540[/tex]
The 95% confidence interval for the true proportion of Canadian chefs who believe gluten-free is the most popular culinary theme us (0.2574, 0.3540).
c. Find a 95% confidence interval for the true proportion of Canadian chefs who believe bite-size desserts are the most popular dessert item\
98 selected bite-size desserts as the most popular dessert item, so [tex]\pi = \frac{98}{350} = 0.28[/tex]
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{350}} = 0.28 - 1.96\sqrt{\frac{0.28*0.72}{350}} = 0.233[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{350}} = 0.28 + 1.96\sqrt{\frac{0.28*0.72}{350}} = 0.327[/tex]
The 95% confidence interval for the true proportion of Canadian chefs who believe bite-size desserts are the most popular dessert item is (0.233, 0.327).
