Question

Consider the following scenario: An industrial seller of grass seeds packages its product in 50-pound bags. A customer has recently filed a complained alleging that the bags are underfilled. A production manager randomly samples a batch and measures the following weights:

Weight, (lbs)

45.6     49.5

47.7     46.7

47.6     48.8

50.5     48.6

50.2     51.5

46.9     50.2

47.8     49.9

49.3     49.8

53.1     49.3

49.5     50.1

To determine whether the bags are indeed being underfilled by the machinery, the manager must conduct a test of mean with a significance level α = 0.05.

In a minimum of 175 words, respond to the following:

• State appropriate null (Ho) and alternative (H1) hypotheses.
• What is the critical value if we work with a significant level α = 0.05?
• What is the decision rule?
• Calculate the test statistic.
• Are the bags indeed being underfilled?
• Should machinery be recalibrated?

Answer 1

Calculate the mean and sd of the sample.

sample mean = sum of total terms / number of terms

n= 20

sum of all the terms = 982.6

sample mean = 982.6 / 20 = 49.13

calculate sample standard deviation:

Create the following table:

data	data-mean	(data - mean)2
45.6	-3.53	12.4609
47.7	-1.43	2.0449
47.6	-1.53	2.3409
50.5	1.37	1.8769
50.2	1.07	1.1449
46.9	-2.23	4.9729
47.8	-1.33	1.7689
49.3	0.16999999999999	0.028899999999998
53.1	3.97	15.7609
49.5	0.37	0.1369
49.5	0.37	0.1369
46.7	-2.43	5.9049
48.8	-0.33000000000001	0.1089
48.6	-0.53	0.2809
51.5	2.37	5.6169
50.2	1.07	1.1449
49.9	0.77	0.59289999999999
49.8	0.66999999999999	0.44889999999999
49.3	0.16999999999999	0.028899999999998
50.1	0.97	0.9409

the sum of numbers in the last column = 57.742

Step 1:

Step 1:

Null hypothesis states that grass seeds packet weigh 50 pounds

Ho :   = 50

Ha :

Step 2: Test Statistics

Population mean = = 50

sample size = n = 20

sample mean = = 49.13

sample deviation = s = 1.7433

level of significance  = 0.05

Population is normally distributed. Since the sample size is less than 30 and population standard deviation is not given use T statistics.

Step 3:

This is a left tailed test as the customers are complaning that the bags are undefiled.

The t-critical value for a left-tailed test, for a significance level of α=0.05 tc = −1.729

As the t statistics (-2.232) is in the rejection area we reject the Null hypothesis.  

Are the bags undefiled?

Here as the t statistics is in rejection area we reject the Null hypothesis. i.e. the assumption that the mean weight is 50 pounds is not correct at 0.05 level of significance. So the bags are underfilled as this test.

Should the machinery be recalibrate?

​​​​​​​Yes the machinery should be recalibrated.