Answer :
(a) The probability that a random newborn infant will weigh less than 7.48 lbs is approximately 0.4013. (b) The probability that the mean weight of a simple random sample of 100 newborn infants is less than 7.48 lbs is approximately 0.0062.
Newborn infants' weights are approximately normally distributed with a mean of 7.7 lbs and a standard deviation of 0.88 lbs.
(a) To find the probability that a random newborn infant will weigh less than 7.48 lbs, we need to calculate the z-score and then find the corresponding probability from the standard normal distribution table. The z-score can be calculated using the formula:
z = (x - μ) / σ,
where x is the weight, μ is the mean, and σ is the standard deviation.
For this case, x = 7.48 lbs, μ = 7.7 lbs, and σ = 0.88 lbs.
Plugging these values into the formula, we get
z = (7.48 - 7.7) / 0.88
≈ -0.25.
Using the standard normal distribution table, we find that the probability corresponding to a z-score of -0.25 is approximately 0.4013. Therefore, the probability that a random newborn infant will weigh less than 7.48 lbs is approximately 0.4013.
(b) To find the probability that the mean weight of a simple random sample of 100 newborn infants is less than 7.48 lbs, we can use the Central Limit Theorem. According to the Central Limit Theorem, the sampling distribution of the sample means will be approximately normally distributed, regardless of the shape of the population distribution, as long as the sample size is large enough (typically n ≥ 30). The mean of the sampling distribution of sample means is equal to the population mean, which is 7.7 lbs. The standard deviation of the sampling distribution of sample means (also known as the standard error) can be calculated using the formula:
σ / sqrt(n),
where σ is the population standard deviation and n is the sample size.
For this case, σ = 0.88 lbs and
n = 100.
Plugging these values into the formula, we get the
standard error = 0.88 / sqrt(100)
= 0.088 lbs.
Now, we need to calculate the z-score using the formula:
z = (x - μ) / standard error.
Plugging in x = 7.48 lbs,
μ = 7.7 lbs, and the standard error = 0.088 lbs, we get z = (7.48 - 7.7) / 0.088 ≈ -2.5.
Using the standard normal distribution table, we find that the probability corresponding to a z-score of -2.5 is approximately 0.0062. Therefore, the probability that the mean weight of a simple random sample of 100 newborn infants is less than 7.48 lbs is approximately 0.0062.
(c) To determine what we can conclude from the mean weight of the 112 newborn humans delivered by a certain ObGyn, we need to compare it with the population mean and consider the variability of the sample. The population mean is 7.7 lbs, and the sample mean is 7.45 lbs.
To assess whether the difference between the sample mean and the population mean is statistically significant, we can perform a hypothesis test. We would set up our null hypothesis as the mean weight of newborn infants delivered by this ObGyn being equal to the population mean (7.7 lbs). The alternative hypothesis would be that the mean weight is different from the population mean.By conducting a hypothesis test and calculating the appropriate test statistic, such as a t-test, we can determine if the sample mean is significantly different from the population mean.
Without conducting the test, we cannot draw any specific conclusions about this doctor's medical practice based solely on the mean weight of the newborns. However, if the sample mean consistently deviates significantly from the population mean in further studies, it may suggest that this ObGyn's practice is associated with either higher or lower birth weights. Conducting further investigation would be necessary to make a definitive conclusion.
In summary: (a) The probability that a random newborn infant will weigh less than 7.48 lbs is approximately 0.4013. (b) The probability that the mean weight of a simple random sample of 100 newborn infants is less than 7.48 lbs is approximately 0.0062. (c) No specific conclusions can be made about the doctor's medical practice based solely on the mean weight of the newborns, but further investigation would be necessary if consistent deviations from the population mean are observed.
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