Question

A company packages pretzels into individual bags, each having an advertised mean weight of 454.0 g. A quality control engineer takes regular samples and test them. From experience, the engineer can expect a population standard deviation of 7.8 g when the machinery is operating correctly. During one such test, a random sample of 25 bag shows the mean weight to be 450.0 g. Is there a problem with the equipment at the 5% level of confidence? What would be the maximum deviation between the sample mean weight and the target weight for the machinery to be within specification?

hello I am in a Experimental Measuers class which is similar to statistics and probability and need help with the above question.

Answer 1

Solution

Let

X = weight (g) of pretzels in a bag.

Let mean and standard deviation of X be µ and σ; where σ = 7.8 [given] .................................... (1)

Part (a)

Hypotheses:

Null H₀: µ = µ₀ = 454 Vs Alternative H_A: µ ≠ 454

Test statistic:

Z = (√n)(Xbar - µ₀)/σ,

where

n = sample size;

Xbar = sample average;

σ = known population standard deviation.

Summary of Excel Calculations is given below:

n	25
µ0	454
σ	7.8
Xbar	450
Zcal	-2.5641
Given α	0.05
Zcrit	1.9600
p-value	0.0103

Distribution, Level of Significance, α, Critical Value and p-value

Under H₀, Z ~ N(0, 1)

Critical value = upper (α/2)% point of N(0, 1).

p-value = P(Z > | Zcal |)

Using Excel Functions: Statistical NORMSINV and NORMSDIST, Zcrit and p-value are found to be as shown in the above table.

Decision:

Since | Zcal | > Zcrit, or equivalently, since p-value < α. H₀ is rejected.

Conclusion:

There is sufficient evidence to suggest that there is a problem with the equipment. Answer 1

Part (b)

Machinery is functioning within specification if the above null hypothesis is accepted.

i.e., if | Zcal | < Zcrit

Or, if |{(√n)(Xbar - µ₀)/σ| } < 1.96

Or, if | (Xbar - µ₀) | < (1.96σ/√n)

Or, if | (Xbar - µ₀) | < 3.06 [substituting values of σ and n ]

=> deviation between the sample mean weight and the target weight < 3.06.

Thus, the maximum deviation between the sample mean weight and the target weight for the machinery to be within specification is 3 g Answer 2

DONE