A company that makes shampoo wants to test whether the average amount of shampoo per bottle is 16 ounces. The population standard deviation is known to be 0.20 ounces. Assuming that the hypothesis test is to be performed using 0.10 level of significance and a random sample of n = 64 bottles, how large could the sample mean be before they would reject the null hypothesis? (hint: think about X-bar Alpha.)

Question

BlackTom AI · Accepted Answer

We start by identifying the null hypothesis and the test setup. H0: mu = 16 ounces, with known population standard deviation sigma = 0.20 ounces, sample size n = 64, and significance level alpha = 0.10 for a two-sided test.

First, compute the standard error of the mean: SE = sigma / sqrt(n) = 0.20 / sqrt(64) = 0.20 / 8 = 0.025.

For a two-sided test at alpha = 0.10, the critical z-value is z_{alpha/2} = z_{0.05} ≈ 1.6449. The rejection region is |Xbar - mu0| > z_{alpha/2} * SE.

Calculate the margin of error: ME = z_{alpha/2} * SE ≈ 1.6449 * 0.025 ≈ 0.0411.

Thus, the rejection thresholds for the sample mean are:
- Lower bound: 16 - 0.0411 ≈ 15.9589 ounces
- Upper bound: 16 + 0.0411 ≈ 16.0411 ounces

Any observed Xbar beyond these bounds would lead to rejecting H0 at the 10% level.

Now evaluate the provided options:
- 15.8 ounces: This value is well below the lower tolerance 15.9589; an Xbar this small would certainly lead to rejecting H0, but the question asks how large Xbar could be before rejection, so this is not the relevant threshold.
- 16.98 ounces: This is far above the upper threshold 16.0411; if Xbar were this large, we would reject, but it is not the boundary value that marks the exact cutoff.
- 16.041 ounces: This matches the calculated upper rejection threshold (rounded to three decimals). If Xbar equals or exceeds this value, H0 would be rejected at alpha = 0.10 for a two-sided test.
- 16.78 ounces: This is well beyond the critical boundary and would also lead to rejection, but it is not the precise cutoff used for the boundary determination.

Summary: among the options, 16.041 ounces aligns with the computed upper boundary of the non-rejection region, representing the largest sample mean that would still not cause rejection, with any larger value prompting a rejection.

OSC 2030-009 Fall 2025 Exam 2

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