Question

Consider the trash bag problem. Suppose that an independent laboratory has tested trash bags and has found that no 30-gallon bags that are currently on the market have a mean breaking strength of 50 pounds or more. On the basis of these results, the producer of the new, improved trash bag feels sure that its 30-gallon bag will be the strongest such bag on the market if the new trash bag's mean breaking strength can be shown to be at least 50 pounds. The mean of the sample of 40 trash bag breaking strengths in Table 1.9 is xbar = 50.575. If we let μ denote the mean of the breaking strengths of all possible trash bags of the new type and assume that σ equals 1.65:

a. Calculate 90 percent and 95 percent confidence intervals for μ.

b. Using the 90 percent confidence interval, can we be 90 percent confident that μ is at least 50 pounds? Explain.

c. Using the 95 percent confidence interval, can we be 95 percent confident that μ is at least 50 pounds? Explain.

d. Based on your answers to parts b and c, how convinced are you that the new 30-gallon trash bag is the strongest such bag on the market?

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Answer 1

Solutions:-

we are given with x-bar = 50.575, = 1.65 and n = 40

we know that 100(1- ) = confidence internal for the true mean as.

(x-bar Z/2 ( / √n))

(a):-

thus the 90 % confidence interval for the true mean is :

(x-bar Z0.01/ 2 (/√n))

(50.575 Z0.005(1.65 / √40))

(50.5751.645*(0.261))

(50.5750.429345)

(50.1458,51.0043)

~=(50,51)

Thus the 95 % confidence interval for the true mean is:

(x-bar Z0.05/2(/√n))

(50.575Z0.0025(1.65/√40))

(50.5751.96(0.261))

(50.5750.51156)

(50.064,51.086)

~=(50,51)

(b):-

From part (a) we are 90 % confidence that lies between 50 pounds to 51 pounds.

therefore we are 90 % confidence that is atleast 50 pounds.

(c):-

From part (a) we are 95 % confidence that lies between 50 pounds to 51 pounds.

therefore we are 95 % confidence that is atleast 50 pounds.

(d):-

Yes the producer of the new improved trash bag feels sure that its 30 gallon bag will be strongest such bag on the market is the new trash bags mean breaking strength can be show to be atleast 50 pounds the producer of the new improved trash bag feels sure that its 30 gallons bag will be the strongest such bag on the market is tge new trash bags mean breaking strength can be shown to be atleast 50 pounds.

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