Answer:
(a) Correct option (1).
(b) Correct option (3).
(c) Correct option (2).
Step-by-step explanation:
A one-sample <em>z</em>-test for mean is used to determine whether the claimed value of mean is significant or not. To perform this test we can also use the <em>t</em>-distribution.
The <em>t</em>-test is used if the following conditions are given:
- The population is normally distributed
- The population standard deviation is not known
- The sample selected is large enough.
Similarly the (1 - <em>α</em>)% confidence interval for population mean can be computed either using the <em>z</em>-interval or the <em>t</em>-interval.
The same conditions should be applied to use the <em>t</em>-interval.
(a)
In the first case, Gale wants to determine the likely range of weight of whole coffee beans from her trusty two-tablespoon scoop to create a reliable method of brewing the perfect cup of coffee.
She selects a sample of <em>n</em> = 15 scoops and perform the test.
Since the population standard deviation is not known and Gale wants to know the range of weight of whole coffee beans , she should use a one sample <em>t</em>-confidence interval for a mean.
The correct option is (1).
(b)
In the second case, Tanya wants to determine whether the mean IQ score is at least 112 or not.
She selects a a random sample <em>n</em> = 20 composers and compute the statistic to perform the one-sample test.
Since the population standard deviation is not known, Tanya should use one sample t-test for a mean.
The correct option is (3).
(c)
In the second case, Jason wants to estimate what the balance of his retirement account will be in one month.
He selects the data from 1950 to 2015 and calculates this index has had an average monthly return of 0.69% with a standard deviation of 4.15%.
Now, the sample selected is quite large. The Central limit theorem can be used to approximate the sampling distribution of sample mean.
Then, 
So, Jason can use the one sample z-test for a mean to determine the balance of his retirement account will be in one month.
The correct option is (2).
