Question

Oasis Inc. which produces water wants to analyze the mineral contents (gm) in the bottle of 1 liter water they produce. Especially, they are interested in identifying the magnesium contents, which is one of the most important minerals in the water. The quality control engineer at their production unit randomly selected 7 bottles and then measured the mineral contents in the samples. The results obtained from the samples are as shown in the table below:

Sample#

Magnesium weight (gm)

1

11.3 + ……..=…….

2

12.2

3

11.7+……..= ……..

4

12.9

5

12.5+…….=……….

6

11.7

7

12.6

To obtain the weight for sample 1, sample 3 and sample 5 in the table, follow the instruction below:

If the fourth number of your student ID is in between:

- (0 & 3), then add 0.2 to the weight as shown in the table for samples 1, 3 and 5.

- (4 & 6), then add 0.6 to the weight as shown in the table for samples 1, 3 and 5.

- (7 & 9), then add 0.9 to the weight as shown in the table for samples 1, 3 and 5.

Fourth number is 5

For example, if student ID: 116523, then add 0.6 in the given data for samples 1, 3 and 5:

- Sample 1=11.3+0.6 =11.9

- Sample 3 = 11.7 +0.6 = 12.3

- Sample 5 = 12.5+0.6 = 13.1

a. Assuming that the population standard deviation is similar to the standard deviation obtained from the above samples, identify the bounds at ….. % confidence interval (CI)?

(4.0)

Note: If the last number of your student ID is in between:

- (0 & 2), then put CI as 92%

- (3 & 5), then put CI as 94%

- (6 & 7), then put CI as 96%

- (8 & 9), then put CI as 98%

Last number is 3

For example, if student ID: 116523, then CI = 94%.

b. If only one sided confidence bound is desired by the company at the lower tail, then what will be the lower bound for the date at the CI similar to (a). Discuss the difference in the lower bound as compared to (a) and give reason for the difference using simple figure(s).

(3.0)

c. If the company wants to have an error of less than 0.20 in estimating the magnesium content, then what sample size should the company use in part (b) for the analysis? (2.0)

ID=114985

Answer 1

Given that,

Oasis Inc. which produces water wants to analyze the mineral contents (gm) in the bottle of 1-liter water they produce.

Especially, they are interested in identifying the magnesium contents, which is one of the most important minerals in the water.

The quality control engineer at their production unit randomly selected 7 bottles and then measured the mineral contents in the samples.

The results obtained from the samples are as shown in the table below:

Sample	Magnesium weight (gm)
1	11.3+0.9=12.2
2	12.2
3	11.7+0.9=12.6
4	12.9
5	12.5+0.9=13.4
6	11.7
7	12.6

(a).

Here,

n=7

For the given sample, mean weights are:

=87.6/7

=12.5143

The sample standard deviation is:

S=0.5491

At 92% confidence interval, we will be using a critical value for Z-test because population standard deviation is known (Since it is said in the problem that population standard deviation is similar to sample standard deviation).

So, the critical value is,

Therefore,

The 92% confidence interval is given by:

=(12.151, 12.8776)

(b).

The one-sided confidence interval at the lower tail is:

We have,

=12.5143, S=0.5491 and n=7

So, the one-sided confidence interval at lower tail is,

So, the lower bound is 12.2227.

The lower bound in (a) is less than the lower bound in (b) because the margin of error is reduced in (b). So, this one-sided confidence interval give us a more compact bound.

(c).

The sample for the sample size is:

Hence,

The critical value in (b) =1.4051, S=0.5491

So, the sample size,

=14.8803

=15 (Approximately)

Thus, the company should use a sample size of 15 to get a margin of error of less than 0.20.