Suppose a manufacturer claims that each family-size bag of pretzels sold weighs 12 ounces on average with a standard deviation of 0.8 ounces. A consumer's group decides to test this claim. If a simple random sample of 49 bags of pretzels gives a sample mean of 11.8 ounces, is this significant evidence that the actual weight of a bag of pretzels is different from the manufacturer's claim?
(a) State the hypothesis.
(b) Describe and draw the corresponding sampling distribution.
(a) The null hypothesis (H0) is that the actual weight of a bag of pretzels is equal to the manufacturer's claim, μ = 12 ounces. The alternative hypothesis (Ha) is that the actual weight of a bag of pretzels is different from the manufacturer's claim, μ ≠ 12 ounces.
(b) The sampling distribution will follow a normal distribution since the sample size is large enough (n = 49) and the population standard deviation (σ) is known. The mean of the sampling distribution will be equal to the population mean (μ = 12 ounces), and the standard deviation of the sampling distribution (standard error) can be calculated using the formula σ/√n, where σ is the population standard deviation and n is the sample size. In this case, the standard error is 0.8/√49 = 0.8/7 = 0.114 ounces.
The sampling distribution can be represented by a bell-shaped curve with the mean at 12 ounces and a standard deviation of 0.114 ounces.
(a) State the hypothesis:
Null Hypothesis (H0): The actual weight of a bag of pretzels is equal to the manufacturer's claim of 12 ounces.
Alternative Hypothesis (Ha): The actual weight of a bag of pretzels is different from the manufacturer's claim of 12 ounces.
(b) Describe and draw the corresponding sampling distribution:
To analyze the hypothesis test, we can use the sampling distribution of the sample mean. The characteristics of this sampling distribution can be described using the Central Limit Theorem.
Given that the manufacturer claims the average weight of a bag of pretzels to be 12 ounces, the mean of the sampling distribution would also be 12 ounces. The standard deviation of the sampling distribution, commonly known as the standard error, can be calculated using the formula:
Standard Error = (Standard Deviation of the Population) / sqrt(sample size)
In this case, the standard deviation of the population is 0.8 ounces, and the sample size is 49 bags of pretzels. Therefore, the standard error would be:
Standard Error = 0.8 / sqrt(49) = 0.8 / 7 = 0.1143
We can now draw the corresponding sampling distribution, which would be a normal distribution with a mean of 12 ounces and a standard error of 0.1143. The distribution would have a bell-shaped curve centered at 12 ounces.
The observed sample mean of 11.8 ounces can be marked on this sampling distribution.
I'll get you started.
Hypotheses:
Ho: µ = 12 --> this is the null hypothesis
Ha: µ ≠ 12 ---> this is the alternate or alternative hypothesis
Note:
Null hypothesis always uses an equals sign. The alternate or alternative hypothesis in this case uses a "does not equal" sign because the problem is just asking if there is a difference. There is no specific direction mentioned (such as "greater than" or "less than" or similar statements).
I'll let you take it from here.