to estimate p, the proportion of all newborn babies who are male, the gender of 10,000 newborn babies was noted. if 5106 were male determine a) a 90 percent confidence interval estimate of p b) a 99 percent confidence interval estimate of p?

A scratch-card costs 5¤ and has two possible prizes. There is a 10% chance you get 5¤ back

and
a 1% chance you get 100¤. In all other cases, you get nothing. Let X denote your net winnings,
i.e, (X = prize - cost). What are the expectation and standard deviation of X?

1200 and 200

To estimate the proportion of all newborn babies who are male, we can use the sample proportion obtained from the 10,000 newborn babies and construct a confidence interval. Let's calculate the confidence intervals using both the formulas for a 90% and a 99% confidence level.

a) 90 percent confidence interval estimate of p:

To construct a confidence interval at a 90% confidence level, we can use the following formula:

CI = p̂ ± Z * sqrt((p̂ * (1 - p̂)) / n)

where:
p̂ is the sample proportion (male babies / total babies)
Z is the critical value from the standard normal distribution corresponding to the desired confidence level (90% confidence level corresponds to Z ≈ 1.645)
n is the sample size (10,000)

First, calculate the sample proportion:
p̂ = (number of male babies) / (total number of babies) = 5106 / 10000 = 0.5106

Next, calculate the standard error:
SE = sqrt((p̂ * (1 - p̂)) / n) = sqrt((0.5106 * (1 - 0.5106)) / 10000) ≈ 0.0071

Now, calculate the lower and upper bounds of the confidence interval:
Lower Bound = p̂ - (Z * SE) = 0.5106 - (1.645 * 0.0071) ≈ 0.4983
Upper Bound = p̂ + (Z * SE) = 0.5106 + (1.645 * 0.0071) ≈ 0.5229

Therefore, the 90% confidence interval estimate of p is approximately 0.4983 to 0.5229.

b) 99 percent confidence interval estimate of p:

To construct a confidence interval at a 99% confidence level, we follow the same steps as before, but with a different critical value. For a 99% confidence level, Z ≈ 2.576 (taken from the standard normal distribution).

Using the same formula and calculations as in part a), we get:

Lower Bound = p̂ - (Z * SE) = 0.5106 - (2.576 * 0.0071) ≈ 0.4889
Upper Bound = p̂ + (Z * SE) = 0.5106 + (2.576 * 0.0071) ≈ 0.5323

Therefore, the 99% confidence interval estimate of p is approximately 0.4889 to 0.5323.