Question

A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 403 gram setting. It is believed that the machine is underfilling the bags. A 38 bag sample had a mean of 400 grams. Assume the population standard deviation is known to be 11. Is there sufficient evidence at the 0.01 level that the bags are underfilled?

Step 1 of 6: State the null and alternative hypotheses.

Step 2 of 6: Find the value of the test statistic. Round your answer to two decimal places.

Step 3 of 6: Specify if the test is one-tailed or two-tailed.

Step 4 of 6: Find the P-value of the test statistic. Round your answer to four decimal places.

Step 5 of 6: Identify the level of significance for the hypothesis test.

Step 6 of 6: Make the decision to reject or fail to reject the null hypothesis.

Answer 1

Solution:

Step 1 :

The null and alternative hypotheses are as follows:

H_{​​​​​​0} : μ = 403 grams i.e. The population mean of the weights of the bags filled by bag filling machine is 403 grams.

H_{​​​​​​0} : μ < 403 grams i.e. The population mean of the weights of the bags filled by bag filling machine is less than 403 grams.

Step 2 :

To test hypothesis we shall use z-test for single mean. The test statistic is given as follows:

Where, x̄ is sample mean, μ is hypothesized value of population mean, σ is population standard deviation and n is sample size.

We have, x̄ = 400 grams, μ = 403 grams, σ = 11 grams and n = 38

On rounding to 2 decimal places we get, Z = -1.68.

The value of the test statistic is -1.68.

Step 3:

Our test is one-tailed (left-tailed) test.

Step 4 :

Since, our test is left-tailed test, therefore we shall obtain left-tailed p-value for the test statistic. The left-tailed p-value for the test statistic is given as follows:

p-value = P(Z < value of the test statistic)

p-value = P(Z < -1.6812)

p-value = 0.0464

Step 5 :

The level of significance for the hypothesis test is 0.01.

Step 6:

We make decision rule as follows:

If p-value is greater than the significance level, then we fail to reject the null hypothesis (H_{​​​​0}) at given significance level.

If p-value is less than the significance level, then we reject the null hypothesis (H_{​​​​0}) at given significance level

We have, p-value = 0.0464 and significance level = 0.01

(0.0464 > 0.01)

Since, p-value is greater than the significance level of 0.01, therefore we shall be fail to reject the null hypothesis (H_{​​​​​​0}) at significance level of 0.01.

Conclusion: At significance level of 0.01, there is not enough evidence to support the claim that machine is underfilling the bags.

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