A market research firm is studying customer satisfaction for a food delivery service. Based on past data, $85\%$ of customers rate the service as "satisfactory". The firm randomly surveys groups of 250 customers. Find the mean ${\mu }_{\hat{p}}$ and standard deviation ${\sigma }_{\hat{p}}$ of the sampling distribution for $\hat{p}$. What would the shape of the sampling distribution be?

• A. ${\mu }_{\hat{p}}=0.34$${\sigma }_{\hat{p}}=0.0300$The sampling distribution is normal
• B. $\mu_{p̂}=0.85$$\sigma_{p̂}=0.0226$The sampling distribution is skewed
• C. $\mu_{p̂}=0.85$$\sigma_{p̂}=0.0226$The sampling distribution is normal
• D. $\mu_{p̂}=0.34$$\sigma_{p̂}=0.0300$The sampling distribution is skewed

Answer: (C)

Verified step by step guidance

1. Identify the given probability of success (p) as 0.85, which represents the proportion of customers who rate the service as 'satisfactory'.
2. Determine the sample size (n), which is 250 customers in each surveyed group.
3. Calculate the mean of the sampling distribution of the sample proportion (μ_{p̂}) using the formula: μ_{p̂} = p. Substitute the given value of p to find μ_{p̂}.
4. Calculate the standard deviation of the sampling distribution of the sample proportion (σ_{p̂}) using the formula: σ_{p̂} = sqrt((p * (1 - p)) / n). Substitute the given values of p and n to find σ_{p̂}.
5. Determine the shape of the sampling distribution. Since the sample size is large and p is not too close to 0 or 1, the sampling distribution of the sample proportion will be approximately normal according to the Central Limit Theorem.