Observations Samples: What Are They?

The observational sample consisted of 30 bottles of soda containing ounces of product. The mean, median, and standard deviations were determined to better comprehend the required quantity of ounces as advertised by the company.

Mean Solution

14.23 + 14.32 + 14.98 + 15 + 15.11 + 15.21 + 15.42 + 15.47 + 15.65 + 15.74 + 15.77 + 15.8 + 15.82 + 15.87 + 15.98 + 16 + 16.02 + 16.05 + 16.21 + 16.21 + 16.23 + 16.25 + 16.31 + 16.32 + 16.34 + 16.46 + 16.47 + 16.51 + 16.91 + 16.96

475.62

Therefore, the sample mean is 15.854 ounces per soda bottle.

Median

The median is also a measure of central tendency which provides information about the middle observation in the sample. Since the number of observations is even, the middle number will be between the 15th and the 16th observation.

Solution

Median

Standard Deviation

Bottle number

x

Bottle number

x

14.23

-1.624

2.637376

16

0.146

0.021316

14.32

-1.534

2.353156

16.02

0.166

0.027556

14.98

-0.874

0.763876

16.05

0.196

0.038416

15

-0.854

0.729316

16.21

0.356

0.126736

15.11

-0.744

0.553536

16.21

0.356

0.126736

15.21

-0.644

0.414736

16.23

0.376

0.141376

15.42

-0.434

0.188356

16.25

0.396

0.156816

15.47

-0.384

0.147456

16.31

0.456

0.207936

15.65

-0.204

0.041616

16.32

0.466

0.217156

15.74

-0.114

0.012996

16.34

0.486

0.236196

15.77

-0.084

0.007056

16.46

0.606

0.367236

15.8

-0.054

0.002916

16.47

0.616

0.379456

15.82

-0.034

0.001156

16.51

0.656

0.430336

15.87

0.016

0.000256

16.91

1.056

1.115136

15.98

0.126

0.015876

16.96

1.106

1.223236

Solution

Question 2

The 95% confidence interval implies that the level of significance (alpha) is 5%. The t-test was used to calculate the 95% confidence interval at 5% level of significance and (n-1) degrees of freedom. Therefore, the confidence interval is given by;

Student t-test

= 15.854

s = 0.6614

n = 30

The lower limit is given by 15.6071 while the upper limit is 16.109. This represents the range in which the amount of ounces in the soda bottles would lie in the case of repeated sampling.

Question 3

Hypotheses:

Ho: On average the bottles contain 16 ounces of products

H1: On average the bottles contain less than 16 ounces of product.

We proceed to test the validity of the hypotheses using the t-test at 5% level of significance and (n-1) degrees of freedom as shown below.

.Solution

The t calculated value =

We compare the t calculated value with the t critical value obtained from the statistical tables. The t tabulated value is given by;

T tabulated =

In this case, the tabulated t value is smaller than the t calculated value implying that we fail to reject the null hypothesis. Therefore, from the statistical analysis above, the customers’ claim of the bottles containing less than 16ounces of product is not valid.

Question 4

The conclusion of rejecting the customers’ claim is supported statistically at the 95% confidence interval. The upper and the lower bounds provide a narrow range over which the mean number of ounces should lie. In this regard, it is evident that some of the bottles will have number of ounces falling outside the range representing defective figure. The existence of the defectives is an indication of poor quality product produced by the company due to various reasons such as the use of inefficient machineries. To avoid complaints from the customers, the company should check for defectives in the sodas produced.