Answer to Question #258852 in Statistics and Probability for boo

Question

A manufacturer of dog food is concerned about the low levels of sales recorded of one of its tinned

products and wants to consider removing this product from its line. Before doing so, a limited scale

market surveyed of 1000 customers that buy this particular brand of dog food is conducted. Of the

people surveyed, 230 said that they would like the manufacturer to keep the product under

investigation as one of the products in the manufacturer’s line. Use this information to construct the

95% confidence interval for the proportion of all customers of this manufacturing brand that would

like the manufacturer to keep the product under investigation as one of the products in the

manufacturer’s line.

Answer 1

The sample proportion is computed as follows, based on the sample size $N = 1000$ and the number of favorable cases $X = 230$

$$\hat{p}=\dfrac{X}{N}=\dfrac{230}{1000}=0.23$$

The critical value for $\alpha = 0.05$ is $z_c = z_{1-\alpha/2} = 1.96.$

The corresponding confidence interval is computed as shown below:

$$CI(proportion)=(\hat{p}-z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{N}},$$

$$\hat{p}+z_c\sqrt{\dfrac{\hat{p}(1-\hat{p})}{N}})$$

$$=(0.23-1.96\sqrt{\dfrac{0.23(1-0.23)}{1000}},$$

$$0.23+1.96\sqrt{\dfrac{0.23(1-0.23)}{1000}})$$

$$=(0.204,0.256)$$

Therefore, based on the data provided, the 95% confidence interval for the population proportion is $0.204 < p < 0.256,$ which indicates that we are 95% confident that the true population proportion $p$ is contained by the interval $(0.204, 0.256).$