Question

a). A student read that a 95% confidence interval for the mean SAT Mathematics score of Texas high school seniors in 2019 is 467 to 489. Asked to explain the meaning of this interval, the student responded, “In 2019, 95% of Texas high school seniors had SAT Mathematics scores between 467 and 489.” Is the student essentially correct? Justify your answer fully.

b). A consumer group chose to study the true population mean salary (??) of “family practice” doctors in the Greater Houston Metroplex. A random sample of one hundred family practice doctors working in this area produced the 95% confidence interval [$241,150, $249,070] for ??. Answer the following:

i). If possible, find the sample average salary for the one hundred family practice doctors involved in this study. If not possible, then state why this statistic cannot be found.

ii). Find the margin of error associated with this confidence interval.

Answer 1

a)

A student read that a 95% confidence interval for the mean SAT Mathematics score of Texas high school seniors in 2019 is 467 to 489. Asked to explain the meaning of this interval, the student responded, “In 2019, 95% of Texas high school seniors had SAT Mathematics scores between 467 and 489.” Is the student essentially correct?

the student responded,

“In 2019, 95% of Texas high school seniors had SAT Mathematics scores between 467 and 489.”

it means that out of 100 the 95 texas high school seniors had SAT Mathematics scores between 467 and 489.

but it is not true.

It is correct to say that there is a 95% chance that the confidence interval you calculated contains the true population means.

It is not quite correct to say that there is a 95% chance that the population means lies within the interval.

What's the difference?

The population mean has one value. You don't know what it is (unless you are doing simulations) but it has one value.

If you repeated the experiment, that value wouldn't change (and you still wouldn't know what it is). Therefore it isn't strictly correct to ask about the probability that the population mean lies within a certain range.

In contrast, the confidence interval you compute depends on the data you happened to collect. If you repeated the experiment, your confidence interval would almost certainly be different. So it is OK to ask about the probability that the interval contains the population mean.

It is not quite correct to ask about the probability that the interval contains the population mean.

It either does or it doesn't. There is no chance about it. What you can say is that if you perform this kind of experiment many times, the confidence intervals would not all be the same, you would expect 95% of them to contain the population mean, you would expect 5% of the confidence intervals to not include the population mean, and that you would never know whether the interval from a particular experiment contained the population mean or not.

the right interpretation is :

There is a 95% chance that the confidence interval (467 to 489) contains the true population means SAT Mathematics score of Texas high school seniors in 2019.

b).

A consumer group chose to study the true population mean salary (?) of “family practice” doctors in the Greater Houston Metroplex. A random sample of one hundred family practice doctors working in this area produced the 95% confidence interval [$241,150, $249,070] for ?.

i)  sample average salary for the one hundred family practice doctors involved in this study.

xbar = ?

Let margin of error = ME

The confidence interval calculated as ( xbar - ME ; xbar +ME)

Here given :- ( xbar - ME ; xbar +ME) = ( 241150 , 249070)

(xbar - ME ) + ( xbar + ME) = 241150 + 249070

xbar - ME + xbar +ME = 490220

2*xbar = 490220

xbar = 490220/ 2

xbar = 245110

Answer:- 245110

ii) margin of error=?

ME = ?

from confidence interval

Xbar + ME = 249070

245110 + ME = 249070

ME = 249070 - 245110

ME= 3960

Answer:- 3960