Question

The producer of a weight-loss pill advertises that people who use the pill lose, after one week, an average (mean) of 1.75 pounds with a standard deviation of 1.02 pounds. In a recent study, a group of 55 people who used this pill were interviewed. The study revealed that these people lost a mean of 1.77 pounds after one week. If the producer's claim is correct, what is the probability that the mean weight loss after one week on this pill for a random sample of 55 individuals will be 1.77 pounds or more

Answer 1

44.23% probability that the mean weight loss after one week on this pill for a random sample of 55 individuals will be 1.77 pounds or more

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`;

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

`\mu = 1.75, \sigma = 1.02, n = 55, s = (1.02)/(√(55)) = 0.1375`

What is the probability that the mean weight loss after one week on this pill for a random sample of 55 individuals will be 1.77 pounds or more

This is 1 subtracted by the pvalue of Z when X = 1.77. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (1.77 - 1.75)/(0.1375)`

`Z = 0.145`

`Z = 0.145` has a pvalue of 0.5577

1 - 0.5577 = 0.4423

44.23% probability that the mean weight loss after one week on this pill for a random sample of 55 individuals will be 1.77 pounds or more