Question

A manufacturer of salad dressings uses machines to dispense liquid ingredients into bottles that move along a filling line.

The machine that dispenses dressing is working properly when 8 ounces are dispensed. The standard deviation of the process is 0.15 ounces.
A sample of 48 bottles is selected periodically, and the filling line is stopped if there is evidence that the mean amount dispensed is different from 8 ounces.

Suppose that the mean amount dispensed in a particular sample of 48 bottles is 7.983 ounces.

Calculate the P-Value.

Answer 1

To calculate the P-Value, we need to perform a hypothesis test. The test statistic is calculated by subtracting the hypothesized mean from the sample mean and dividing by the standard error. The P-Value is the probability of observing a test statistic as extreme as the calculated value or more extreme under the null hypothesis.

Step-by-step explanation:

To calculate the P-Value, we need to perform a hypothesis test. The null hypothesis, H0, is that the mean amount dispensed is equal to 8 ounces, while the alternative hypothesis, Ha, is that the mean amount dispensed is different from 8 ounces.

Using the given information, we can calculate the test statistic, which is the standardized value of the sample mean. First, we calculate the standard error of the sample mean, which is the standard deviation divided by the square root of the sample size. In this case, the standard error is 0.15 / sqrt(48) = 0.021.

The test statistic is then calculated by subtracting the hypothesized mean (8 ounces) from the sample mean (7.983 ounces) and dividing by the standard error. So, the test statistic is (7.983 - 8) / 0.021 = -0.619.

To calculate the P-Value, we need to find the probability of observing a test statistic as extreme as -0.619 (or more extreme) under the null hypothesis. We can use a standard normal distribution table or a calculator to find the corresponding area under the curve. From the table or calculator, we find that the area to the left of -0.619 is 0.269.

Since the alternative hypothesis is two-sided (the mean could be either greater or less than 8 ounces), we double the area to get the P-Value. So, the P-Value is 2 * 0.269 = 0.538.

Answer 2

P-value = 0.4324

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 8 ounces

Sample mean,
`\bar{x}` = 7.983 ounces

Sample size, n = 48

Population standard deviation, σ = 0.15 ounces

First, we design the null and the alternate hypothesis

`H_(0): \mu = 8\text{ ounces}\\H_A: \mu \\eq 8\text{ ounces}`

We use Two-tailed z test to perform this hypothesis.

Formula:

`z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }`

Putting all the values, we have

`z_(stat) = \displaystyle(7.983 - 8)/((0.15)/(√(48)) ) = -0.7851`

Now, we calculate the p-value with the help of standard normal z table.

P-value = 0.4324