Question

Medicare Hospital Insurance The average yearly Medicare Hospital Insurance benefit per person was $4064

in a recent year. Suppose the benefits are normally distributed with a standard deviation of $460

. Round the final answers to at least four decimal places and intermediate z-value calculations to two decimal places.

Part 1 of 2

Find the probability that the mean benefit for a random sample of 26 patients is less than

$3920

.

=P<X3920

Part 2 of 2

Find the probability that the mean benefit for a random sample of 26 patients is more than

$4270

.

=P>X4270

Answer 1

Solution:

Given that , X follows normal distribution with

= 4064

= 460

A sample of size n = 26 is taken from this population.

Let be the mean of sample.

Since population is normally distributed , the sampling distribution of the is approximately normal with

Mean = = 4064

SD =      = 460/​26 =  90.2134221636

1)

P( < 3920)

= P[( - )/ < (3920 - 4064)/90.2134221636]

= P[Z < -1.60]

= 0.0548

2)

P( > 4270)

= P[( - )/ >  (4270 - )/]

= P[ ( - )/ > (4270 - 4064)/ 90.2134221636]

= P[Z > 2.28]

= 1 - P[Z < 2.28]

= 1 - 0.9887 ( use z table)

= 0.0113