Final answer:
To calculate the probability that the sample proportion is between two values, we would normally use the z-score and standard normal distribution, but this requires the sampling distribution to be normal, which often assumes a sufficiently large sample size where both np and nq are greater than five. If the normality condition is not met, we cannot assume a zero probability; instead, we must use an alternate method such as the binomial distribution or numerical approximations.
Step-by-step explanation:
To determine the probability that the sample proportion is between 0.22 and 0.24 for a normally distributed sampling distribution with n = 13, we need first to verify the normality condition. In general, for sample proportions, we can approximate the distribution as normal if both np and nq are greater than five. To calculate this probability, we would normally use the mean of the sampling distribution p and standard deviation √npq to find the corresponding z-scores and then use the standard normal distribution to find the area between these z-scores which gives us the probability.
In this specific case, however, as n = 13 is quite small, and we are not given p to verify np and nq are greater than five, we cannot assume normality. If the conditions for normality are not met, we cannot simply put the probability as zero; instead, we would need to apply a different distribution that fits smaller samples, such as the binomial or another applicable distribution depending on the context.
To calculate the probability for non-normally distributed sample proportions, the exact distribution of the sample proportion should be used; if it is indeed binomial with a known population proportion, p, we could potentially use the binomial distribution directly with the formula P(X = k) = (^n_k)p^k(1-p)^(n-k) where k is the number of successes in n trials. Alternatively, if the sample size and the success probability don't allow for such calculations directly, simulation or numerical methods may be used to approximate the desired probability.