problems for one - sample proportion z - test...
problems for one - sample proportion z - test\n1. a political party claims that 60% of voters support their candidate. in a random survey of 200 voters, 120 voters express support for the candidate.\n2. a factory claims that 5% of its products are defective. in a sample of 500 products, 25 are found to be defective.\n3. a restaurant claims that 85% of its customers are satisfied with the service. a random sample of 100 customers finds that 80 customers are satisfied.
Answer
# Explanation:
## Step1: Identify the problem type
This is a one - sample proportion z - test problem.
## Step2: For the first example
The claimed proportion $p_0 = 0.6$, sample size $n = 200$, and number of successes $x = 120$. The sample proportion $\hat{p}=\frac{x}{n}=\frac{120}{200}=0.6$.
The z - statistic for one - sample proportion test is $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}$. Substituting the values, we get $z=\frac{0.6 - 0.6}{\sqrt{\frac{0.6\times(1 - 0.6)}{200}}}=0$.
## Step3: For the second example
The claimed proportion $p_0 = 0.05$, sample size $n = 500$, and number of successes (defective products) $x = 25$. The sample proportion $\hat{p}=\frac{x}{n}=\frac{25}{500}=0.05$.
The z - statistic $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}=\frac{0.05 - 0.05}{\sqrt{\frac{0.05\times(1 - 0.05)}{500}}}=0$.
## Step4: For the third example
The claimed proportion $p_0 = 0.85$, sample size $n = 100$, and number of successes (satisfied customers) $x = 80$. The sample proportion $\hat{p}=\frac{x}{n}=\frac{80}{100}=0.8$.
The z - statistic $z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}=\frac{0.8 - 0.85}{\sqrt{\frac{0.85\times(1 - 0.85)}{100}}}\approx\frac{- 0.05}{\sqrt{\frac{0.85\times0.15}{100}}}\approx\frac{-0.05}{\sqrt{0.001275}}\approx - 1.41$.
# Answer:
For the first example: z - statistic = 0; For the second example: z - statistic = 0; For the third example: z - statistic $\approx - 1.41$