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The normal approximation is a method used to estimate probabilities for a binomial distribution when the sample size is large enough. It allows us to replace the binomial distribution with a normal distribution, which is easier to work with mathematically.

In the given problem, there are 40,000 travel agencies, and we are interested in the number of American Society of Travel Agents (ASTA) members within a sample of 5,000 agencies. The proportion of ASTA members is given by the ratio of ASTA members to total agencies, which is 0.275.
This is a fair number for statistical calculations.

For normal approximation to be valid, both the expected number of successes, given by \(np\), and the expected number of failures, given by \(n(1-p)\), should each be greater than 5.
Here, \(np = 5000 \times 0.275 = 1375\) and \(n(1-p) = 5000 \times 0.725 = 3625\), which satisfy the condition.

Thus, we can use the normal distribution to approximate probabilities, such as the probability of finding between 1,200 and 1,400 ASTA members.

Mean and standard deviation are fundamental concepts in statistics that summarize the central tendency and variability of a data set, respectively. In the context of a binomial distribution, these values can be calculated directly from the sample size and the probability of success.

To find the mean (\(\mu\)) of a binomial distribution with parameters \(n\) (sample size) and \(p\) (probability of success), we use the formula: \(\mu = np\).
For the standard deviation (\(\sigma\)), the formula is \(\sigma = \sqrt{np(1-p)}\). In the problem, for a sample size of 100 agencies with a probability \(p = 0.275\), the mean is \(27.5\) ASTA members because \(np = 100 \times 0.275 = 27.5\).

The standard deviation can be computed as \(\sigma = \sqrt{100 \times 0.275 \times 0.725} \approx 4.4295\).
This provides insight into how much variation exists from the mean number of ASTA members in a sample of 100 agencies.

The effect of sample size on standard deviation is crucial in understanding distribution properties. The standard deviation indicates how much individual data points deviate on average from the mean.

According to the standard deviation formula \(\sigma = \sqrt{np(1 - p)}\), the standard deviation depends on both the sample size \(n\) and the success probability \(p\).
When the sample size doubles, the new standard deviation is not simply twice the original value. Instead, it increases by the factor of \(\sqrt{2}\). This happens because \(\sigma_{new} = \sqrt{2np(1 - p)} = \sqrt{2} \cdot \sqrt{np(1 - p)}\).

This shows that while the sample size has a direct impact on standard deviation, the relationship is not linear. Doubling the sample size increases variability, but not to the same degree as the increase in sample size itself. This understanding helps in planning studies and ensuring adequate sample sizes are chosen for desired confidence in results.