Questions: The length of human pregnancies is approximately normal with mean μ=266 days and standard deviation σ=16 days. Complete parts (a) through ( f ) A. If 100 independent random samples of size n=25 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of exactly 258 days. B. If 100 independent random samples of size n=25 pregnancies were obtained from this population, we would expect 1 sample(s) to have a sample mean of 258 days or less. C. If 100 independent random samples of size n=25 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of 258 days or more. (e) What might you conclude if a random sample of 25 pregnancies resulted in a mean gestation period of 258 days or less? This result would be unusual, so the sample likely came from a population whose mean gestation period is less than 266 days. (f) What is the probability a random sample of size 19 will have a mean gestation period within 12 days of the mean? The probability that a random sample of size 19 will have a mean gestation period within 12 days of the mean is (Round to four decimal places as needed.)

The length of human pregnancies is approximately normal with mean μ=266 days and standard deviation σ=16 days. Complete parts (a) through ( f ) A. If 100 independent random samples of size n=25 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of exactly 258 days. B. If 100 independent random samples of size n=25 pregnancies were obtained from this population, we would expect 1 sample(s) to have a sample mean of 258 days or less. C. If 100 independent random samples of size n=25 pregnancies were obtained from this population, we would expect sample(s) to have a sample mean of 258 days or more. (e) What might you conclude if a random sample of 25 pregnancies resulted in a mean gestation period of 258 days or less? This result would be unusual, so the sample likely came from a population whose mean gestation period is less than 266 days. (f) What is the probability a random sample of size 19 will have a mean gestation period within 12 days of the mean? The probability that a random sample of size 19 will have a mean gestation period within 12 days of the mean is (Round to four decimal places as needed.)
Transcript text: The length of human pregnancies is approximately normal with mean $\mu=266$ days and standard deviation $\sigma=16$ days. Complete parts (a) through ( f ) A. If 100 independent random samples of size $n=25$ pregnancies were obtained from this population, we would expect $\square$ sample(s) to have a sample mean of exactly 258 days. B. If 100 independent random samples of size $n=25$ pregnancies were obtained from this population, we would expect $\square$ 1 sam sample(s) to have a sample mean of 258 days or less. C. If 100 independent random samples of size $n=25$ pregnancies were obtained from this population, we would expect $\square$ sample(s) to have a sample mean of 258 days or more. (e) What might you conclude if a random sample of 25 pregnancies resulted in a mean gestation period of 258 days or less? This result would be unusual, so the sample likely came from a population whose mean gestation period is less than 266 days. (f) What is the probability a random sample of size 19 will have a mean gestation period within 12 days of the mean? The probability that a random sample of size 19 will have a mean gestation period within 12 days of the mean is $\square$ (Round to four decimal places as needed.)
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Solution

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Solution Steps

Step 1: Expected Number of Samples with Mean Exactly 258 Days

The probability of a continuous random variable taking an exact value is \(0\). Therefore, the expected number of samples with a mean of exactly \(258\) days is:

\[ \text{Expected number of samples} = 0 \]

Step 2: Probability of Sample Mean Being 258 Days or Less

To find the probability that the sample mean is \(258\) days or less, we calculate:

\[ P(X \leq 258) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-2.5) - \Phi(-\infty) = 0.0062 \]

Thus, the expected number of samples with a mean of \(258\) days or less is:

\[ \text{Expected number of samples} = 100 \times 0.0062 = 0.62 \]

Step 3: Probability of Sample Mean Being 258 Days or More

To find the probability that the sample mean is \(258\) days or more, we calculate:

\[ P(X \geq 258) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(-2.5) = 0.9938 \]

Thus, the expected number of samples with a mean of \(258\) days or more is:

\[ \text{Expected number of samples} = 100 \times 0.9938 = 99.38 \]

Final Answer

  • Expected number of samples with mean exactly \(258\) days: \(\boxed{0}\)
  • Expected number of samples with mean \(258\) days or less: \(\boxed{0.62}\)
  • Expected number of samples with mean \(258\) days or more: \(\boxed{99.38}\)
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