The birth of a baby is considered to be premature if the baby is born before 37 weeks of pregnancy have
been completed. According to a study, 10% of babies born in the United States in a particular year were
premature. Suppose that a random sample of 58 babies born in the U.S. in this year will be selected. The
proportion of the babies who were premature, p̂, will be calculated.
(a) What are the mean and standard deviation of the sampling distribution of p̂? (Round your standard
deviation to four decimal places.)
mean
standard deviation
(b) Is the sampling distribution of p̂ approximately normal? Explain
The sampling distribution of p̂ is approximately normal because np and n(1 − p) are both at
least 10.
The sampling distribution of p̂ is approximately normal because np and n(1 − p) are both less
than 10.
The sampling distribution of p̂ is not approximately normal because n(1 − p) is at least 10.
The sampling distribution of p̂ is not approximately normal because n(1 − p) is less than 10.
The sampling distribution of p̂ is not approximately normal because np is less than 10.
(c) What is the smallest value of n for which the sampling distribution of p̂ would be approximately
normal?
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(a) The mean of the sampling distribution of p̂ is equal to the population proportion, which is given as 0.1 (or 10% in decimal form).
The standard deviation of the sampling distribution of p̂ can be calculated using the formula:
Standard deviation = sqrt( (p * (1-p))/n )
Where p is the population proportion (0.1) and n is the sample size (58).
Plugging in the values:
Standard deviation = sqrt( (0.1 * (1-0.1))/58 ) ≈ 0.0433
So, the mean is 0.1 and the standard deviation is 0.0433.
(b) The sampling distribution of p̂ is approximately normal because np and n(1 − p) are both at least 10. In this case, np = 58 * 0.1 = 5.8 and n(1 − p) = 58 * (1 - 0.1) = 52.2. Since both values are greater than 10, the sampling distribution of p̂ is approximately normal.
(c) The smallest value of n for which the sampling distribution of p̂ would be approximately normal is when both np and n(1 − p) are at least 10. In this case, np and n(1 − p) should both be greater than or equal to 10.
Let's solve for n:
np ≥ 10
58 * 0.1 ≥ 10
5.8 ≥ 10
The inequality is not satisfied, so n = 58 is the smallest value for which the sampling distribution of p̂ would be approximately normal.
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