For $n=49$ and $p=0.5$, check the binomial distribution to see whether it can be approximated by a normal distribution.
To determine if a binomial distribution \(B(n, p)\) can be approximated by a normal distribution, we typically check the success-failure condition. A standard rule of thumb for this approximation is that both the expected number of successes \(np\) and the expected number of failures \(n(1-p)\) must be sufficiently large, usually at least \(5\) or \(10\). In this problem, we are given \(n = 49\) and \(p = 0.5\).
First, we calculate the expected number of successes:
\[np = 49 \times 0.5 = 24.5\]
Next, we calculate the expected number of failures:
\[n(1 - p) = 49 \times (1 - 0.5) = 49 \times 0.5 = 24.5\]
Both calculated values are \(24.5\), which is significantly greater than the common threshold of \(5\).
Since both \(np = 24.5\) and \(n(1-p) = 24.5\) satisfy the requirement of being greater than \(5\), the conditions for the normal approximation are met. Therefore, the binomial distribution with \(n=49\) and \(p=0.5\) can be approximated by a normal distribution.
In statistics, the normal approximation is a powerful tool used to simplify probability calculations for large binomial experiments. For example, if you are analyzing the results of 49 coin flips to predict the likelihood of heads, instead of summing individual binomial probabilities, you can use a normal curve with mean \(\mu = np = 24.5\) and standard deviation \(\sigma = \sqrt{np(1-p)} = \sqrt{49 \times 0.5 \times 0.5} = 3.5\) to get a highly accurate estimate. This approach is essential in fields like quality control and survey sampling, where thousands of trials are often conducted.