At 12 months postpartum, women in the control group gained a...

At 12 months postpartum, women in the control group gained a mean of 7.9 lbs. with a standard deviation of 15.3 lbs. compared with their pre-pregnancy weight.
What is the probability that a control group woman was at or below her pre-pregnancy weight at 12 months?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean. It is often used to model naturally occurring phenomena, with many values clustering around the mean and tapering off as they move away. In this context, the weight differences of women are assumed to follow a normal distribution, which allows for probabilistic predictions about individual weight differences.

Mean and Standard Deviation

The mean represents the average value of a data set, while the standard deviation measures the dispersion or spread of the data around the mean. In this problem, the mean weight gain and the standard deviation are essential in defining the normal distribution, which in turn is used to calculate the probability associated with a specific weight difference.

Z-score Standardization

Z-score standardization is a technique that transforms a data point into the number of standard deviations it is away from the mean, allowing for the comparison of values from different normal distributions. By converting the weight difference into a z-score, one can use standard normal distribution tables or functions to find the associated cumulative probability.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a normal distribution gives the probability that a normally distributed random variable will take a value less than or equal to a given point. This function is instrumental in calculating the probability in question, as it aggregates the probabilities from the far left of the distribution up to the point of interest.

Transcript

0:00 Hello.

00:01 So here we get the mean for the sample mean, mu sub x bar, is equal to mu, the mean.

00:08 So here it's going to be equal to seven.

00:09 And we're given the standard error of the sample mean, sigma subx bar, is going to be equal to sigma over the square root of n.

00:19 So here it's going to be equal to we're given the probability the newborn baby.

00:23 We have a weight within 0 .6 pounds of the mean.

00:26 So therefore, the standard deviation is going to be 0 .6.

00:30 So sigma is 0 .60 divided by well here n is 1.

00:34 So it's divided by squared or 1 is equal to just 0 .60 so then the probability that the newborn baby will have a weight within 0 .6 pounds of the mean is well, we first we compute the z score to find the probability based in our standard table so our x here is 6 .40.

00:58 So we get that z is equal to to 6 .40 minus 7 divided by sigma sub x bar, divided by 0 .60, which is about here negative, which is negative 1 .00.

01:13 And then we get that using 7 .60, we get that z is equal to 7 .60 minus 7 over 0 .60, which is that equal to 1 .00.

01:33 So then from the standard table, we get the probability that 6 .40 is less than x, which is less than 7 .60, is going to be equal to, while we get the probability that z is less than equal to 1, minus the probability that z is less than equal to negative 1, that's going to give us 0 .8 .413, minus 0 .1587, giving us 0 .6826 or 68%...

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