Question

In: Statistics and Probability

A pollster claims that the mean amount spent on Christmas gifts by an American family is...

A pollster claims that the mean amount spent on Christmas gifts by an American family is
$930 with a standard deviation of $180. What is the probability that the mean amount spent on
Christmas gifts for a sample of 500 families is between $910 and $970?

The distribution of the durations of human pregnancies in the United States is approximately
normal with mean 268 days and standard deviation 14 days.
a) What is the probability that a randomly selected US woman has a pregnancy which
lasts longer than 276 days?

b) What is the probability that an SRS of 20 US pregnancies show a sample mean that is
greater than 276 days?

Solutions

Expert Solution

Given Information:

Let X denotes the amount spent on Christmas gifts by an American family in dollars.

If 500 families are randomly selected then as the sample size is larger than 30 Central Limit Theorem can be easily applied:

The probability that the mean amount spent on Christmas gifts for a sample of 500 families is between 910 and 970 is calculated using standard normal table and is given by:

Hence, the probability that the mean amount spent on Christmas gifts for a sample of 500 families is between 910 and 970 is 0.9934.

Let Y denotes the durations of human pregnancies in the United States.

The probability that a randomly selected US woman has a pregnancy which
lasts longer than 276 days is calculated using standard normal table and is given by:

The probability that a randomly selected US woman has a pregnancy which
lasts longer than 276 days is 0.2843.

If a simple random sample of 20 US pregnancies is taken under consideration.

The probability that an SRS of 20 US pregnancies show a sample mean that is
greater than 276 days is calculated using standard normal table and is given by:

The probability that an SRS of 20 US pregnancies show a sample mean that is
greater than 276 days is 0.0052.

orchestra answered 1 year ago
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