A recent survey found that 70% of all adults over 50 wear glasses for driving. You randomly select 30 adults over 50, and ask if he or she wears glasses. Decide whether you can use the normal distribution to approximate the binomial distribution. If so, find the mean and standard deviation. If not, explain why.

To decide whether we can use the normal distribution to approximate the binomial distribution in this scenario, we need to check whether the conditions for the normal approximation of the binomial distribution are met.

The conditions for using the normal distribution to approximate the binomial distribution include:
1. The number of trials must be large (in this case, random selection of 30 adults).
2. The number of successes (wearing glasses) must be reasonably large (at least 5).

In this case, the number of trials (30 adults selected) is not extremely large, but it can still be sufficient if the number of successes (wearing glasses) is reasonably large. However, the survey indicates that 70% of all adults over 50 wear glasses for driving. This means that the expected number of successes (n*p) is 30 * 0.7 = 21.

Since the expected number of successes is reasonably large (greater than 5), we can use the normal distribution to approximate the binomial distribution.

To find the mean and standard deviation for the normal distribution approximation, we can use the following formulas:
Mean (μ) = n * p
Standard Deviation (σ) = √(n * p * q)

where n is the number of trials, p is the probability of success, and q is the probability of failure (1 - p).

In this case,
n = 30 (number of adults selected)
p = 0.7 (probability of wearing glasses)
q = 1 - p = 0.3 (probability of not wearing glasses)

Calculating the mean:
μ = n * p = 30 * 0.7 = 21

Calculating the standard deviation:
σ = √(n * p * q) = √(30 * 0.7 * 0.3) ≈ √6.3 ≈ 2.51

So, in this scenario, we can use the normal distribution to approximate the binomial distribution, and the mean and standard deviation for the normal approximation are 21 and 2.51, respectively.

To determine whether we can use the normal distribution to approximate the binomial distribution in this case, we need to check if the conditions for using the normal approximation to the binomial hold.

The conditions for using the normal approximation to the binomial are as follows:
1. The number of trials, n, is large (usually greater than or equal to 20)
2. The probability of success, p, for each trial is not extremely close to 0 or 1

In this case, the number of trials is 30 (n = 30), which satisfies the first condition as it is greater than 20. However, we don't know the exact value of the probability of success (p) for each trial. We only know the overall proportion of adults over 50 who wear glasses for driving, which is 70% or 0.7.

So, to use the normal approximation, we need to check whether np and n(1-p) both exceed 5.

np = 30 * 0.7 = 21
n(1-p) = 30 * (1 - 0.7) = 9

Both np and n(1-p) exceed 5, so we can use the normal distribution to approximate the binomial distribution in this case.

To find the mean and standard deviation for the normal distribution, we can use the formulas:

Mean (μ) = np = 30 * 0.7 = 21
Standard Deviation (σ) = sqrt(np(1-p)) = sqrt(30 * 0.7 * 0.3) ≈ 2.09

Therefore, we can approximate the binomial distribution with a normal distribution with a mean of 21 and a standard deviation of approximately 2.09.

Mean = np = 30 * .7 = ?

Standard deviation = √npq = √(30 * .7 * .3) = ?

Note: q = 1 - p

I'll let you finish the calculations.