if the probability of a newborn child being female is 0.5, find the probability that in 100 births, 55 or more will be female.use the normal distribution to approximate the binomial distribution
Mean = np = 100 * 0.5 = ?
Standard deviation = √npq = √(100 * 0.5 * 0.5) = ?
Note: q = 1 - p
Finish the calculations, then use z-scores.
z = (x - mean)/sd
Note: x = 55
Determine the probability using the z-score and a z-table.
I hope this will help get you started.
.1841
To approximate the binomial distribution using the normal distribution, we need to calculate the mean and standard deviation of the binomial distribution first.
The mean (μ) of a binomial distribution is given by: μ = n * p
Where n is the number of trials (100 births) and p is the probability of success (0.5, the probability of a newborn child being female).
So, μ = 100 * 0.5 = 50
The standard deviation (σ) of a binomial distribution is given by: σ = √(n * p * (1 - p))
So, σ = √(100 * 0.5 * (1 - 0.5)) = √(100 * 0.5 * 0.5) = √25 = 5
Now, we can use the normal distribution to approximate the binomial distribution. We need to calculate the z-score for 55 or more female births.
The z-score (z) is given by: z = (x - μ) / σ
Where x is the number of female births (55) and μ is the mean (50) and σ is the standard deviation (5).
So, z = (55 - 50) / 5 = 1
Now, we need to find the probability (P) of z ≥ 1. We can use a standard normal table to find this probability.
Looking up the z-score of 1 in the standard normal table, we find that the probability of z ≤ 1 is approximately 0.8413. Since we are interested in z ≥ 1, we subtract this probability from 1:
P(z ≥ 1) = 1 - 0.8413 = 0.1587
So, the probability that in 100 births, 55 or more will be female is approximately 0.1587 or 15.87%.
To find the probability that 55 or more out of 100 births will be female, we can approximate the binomial distribution using the normal distribution.
The binomial distribution is used to model the probability of a certain number of successes (in this case, female births) in a fixed number of independent trials (in this case, 100 births) when the probability of success (being female) is known.
First, let's calculate the mean and standard deviation of the binomial distribution. The mean of a binomial distribution is given by the formula:
mean = n * p
where n is the number of trials and p is the probability of success. In this case, n = 100 and p = 0.5, so the mean is:
mean = 100 * 0.5 = 50
The standard deviation of a binomial distribution is given by the formula:
standard deviation = sqrt(n * p * (1 - p))
Using the same values for n and p, the standard deviation is:
standard deviation = sqrt(100 * 0.5 * (1 - 0.5)) = sqrt(25) = 5
Now, we can use the normal distribution to approximate the binomial distribution. We need to standardize the values of 55 or more female births using the z-score formula:
z = (x - mean) / standard deviation
For x = 55:
z = (55 - 50) / 5 = 1
To find the probability of 55 or more female births, we can use a standard normal distribution table or use a calculator with the cumulative distribution function (CDF).
Using a standard normal distribution table, we can find the probability corresponding to a z-score of 1. The table shows that the probability corresponding to a z-score of 1 is approximately 0.8413.
So, the probability that in 100 births, 55 or more will be female is approximately 0.8413, or 84.13% (rounded to two decimal places).
Note: The approximation of the binomial distribution using the normal distribution becomes more accurate as the number of trials becomes larger. In this case, with n = 100, the approximation is reasonable.