The proportion of baby boys born in the United States has historically been 0.512. You choose an SRS of 55 newborn babies and find that 47% are boys. Do ALL calculations to 5 decimal places before rounding.
(d) In a random sample of 55 newborn babies, what is the probability that 47% of the sample are boys. (Use 3 decimal places). Hint: you are asked to find P( < .47 ). To calculate this first find the z-score and then look up the answer in the Z table.
Formula for z-score:
z = (x - mean)/sd
Find mean:
mean = np = (55)(0.512) = ?
Find standard deviation:
sd = √npq = √(55)(0.512)(0.488) = ?
Note: q = 1 - p
Finish both calculations.
Substitute mean and standard deviation into the z-formula. Use 25.85 for x (0.47 * 55 = 25.85).
Once you have the z-score, use the z-table to determine your probability.
I hope this will help get you started.
Thank you sooo much!! Also thanks for posting steps and not just an answer.
To find the probability that 47% of the sample are boys in a random sample of 55 newborn babies, we first need to calculate the z-score using the formula:
z = (x - μ) / (σ / sqrt(n))
Where:
x is the observed proportion (47% or 0.47),
μ is the population proportion (0.512),
σ is the standard deviation of the sampling distribution of proportions, and
n is the sample size (55).
The standard deviation of the sampling distribution of proportions can be calculated using the formula:
σ = sqrt((μ * (1 - μ)) / n)
Plugging in the given values, we have:
μ = 0.512
n = 55
Calculating the standard deviation:
σ = sqrt((0.512 * (1 - 0.512)) / 55)
= sqrt(0.248832 / 55)
≈ sqrt(0.004524)
σ ≈ 0.06730 (rounded to 5 decimal places)
Now, we can calculate the z-score:
z = (0.47 - 0.512) / (0.06730 / sqrt(55))
= (0.47 - 0.512) / (0.06730 / 7.4162) (approximating sqrt(55) to 7.4162)
= -0.042 / (0.06730 / 7.4162)
≈ -0.042 / 0.009073
z ≈ -4.6269 (rounded to 5 decimal places)
Now, we can look up the z-score (-4.6269) in the Z-table to find the corresponding probability. However, since the given table only goes up to 3 decimal places, we will use the closest available value, which is -4.63.
From the Z-table, we find that the probability associated with a z-score of -4.63 is approximately 0.0000.
Therefore, the probability that 47% of the sample are boys is approximately 0.000 (rounded to 3 decimal places).
To calculate the probability that 47% of the sample are boys, we need to find the z-score and then look up the answer in the Z table. The z-score measures how many standard deviations away from the mean the observed value is.
First, we need to calculate the standard deviation of the sample proportion. The formula to calculate the standard deviation of a sample proportion is:
σ = √(p*(1-p) / n)
Where:
σ = standard deviation of the sample proportion
p = proportion of boys born in the United States (0.512)
n = sample size (55)
Substituting the values:
σ = √(0.512*(1-0.512) / 55)
≈ √(0.262144 / 55)
≈ √0.004766253
≈ 0.06899 (rounded to 5 decimal places)
To calculate the z-score, we use the formula:
z = (x - μ) / σ
Where:
z = z-score
x = observed value (0.47)
μ = mean value (expected proportion of boys, which is 0.512)
σ = standard deviation of the sample proportion (0.06899)
Substituting the values:
z = (0.47 - 0.512) / 0.06899
= -0.042 / 0.06899
≈ -0.608 (rounded to 3 decimal places)
Now, we can look up the probability associated with the z-score of -0.608 in the Z table. The table will give us the probability from the left tail of the standard normal distribution (P(<z)).
Using a Z table or a statistical calculator, we find that the probability is approximately 0.270 (rounded to 3 decimal places).
Therefore, the probability that 47% of the sample are boys in a random sample of 55 newborn babies is approximately 0.270.