The average checkout time ( 𝜇 ) at a local Safeway grocery store is hypothesized to be 3 minutes or less (null hypothesis H0: 𝜇

Question

The average checkout time (
𝜇
) at a local Safeway grocery store is hypothesized to be 3 minutes or less (null hypothesis H0: 
𝜇
 <=3). A sample of 36 customers at the local store shows an average checkout time of 3.3 minutes with a standard deviation of 0.6 minutes.
Using a one-sided z-test, what is the p-value of the observed z-statistic for testing H0: 
𝜇
 <=3 versus Ha: 
𝜇
 > 3.

BlackTom AI · Accepted Answer

We start by identifying the test setup and the given values. The null hypothesis is H0: mu <= 3, and the alternative is Ha: mu > 3, using a one-sided z-test. The sample size is n = 36, the observed sample mean is xbar = 3.3, and the provided standard deviation is sigma = 0.6. Since this is labeled a z-test, we treat sigma as known and compute the test statistic accordingly.

Compute the z-statistic: z = (xbar - mu0) / (sigma / sqrt(n)) = (3.3 - 3) / (0.6 / sqrt(36)) = 0.3 / (0.6 / 6) = 0.3 / 0.1 = 3.0. A z of 3.0 corresponds to a right-tailed probability (p-value) under the standard normal distribution.

Now evaluate the p-value for z > 3.0. The standard normal CDF at 3.0 is Phi(3.0) ≈ 0.99865, so the upper tail probability is 1 - Phi(3.0) ≈ 0.00135, which rounds to 0.0014 when expressed to four decimal places.

Option-by-option assessment:
- 0.0014: This is the correct p-value approximation for z = 3.0 in a one-sided test, so it accurately reflects the observed evidence against H0 in favor of Ha.
- 0.0035: This value would correspond to a larger tail probability than the correct one; it would be appropriate if z were around 2.5 to 2.6, not 3.0. Using it would overstate the p-value and understate the strength of evidence.
- 0.0024: While closer than 0.0035, it still overestimates the p-value relative to the exact 0.00135 for z = 3.0. It implies weaker evidence than actually observed.
- 0.0052: This is substantially larger than the correct tail probability for z = 3.0 and would indicate much weaker evidence against H0 than what the data support.
- 0.0073: Similar to the previous, this option greatly inflates the p-value, misrepresenting the strength of the observed effect.

In summary, the calculation yields a p-value of about 0.00135, so 0.0014 is the precise match among the options. The other numbers reflect larger tail probabilities and thus indicate weaker evidence against the null hypothesis than what the data provide.

BU.510.601.K1.FA25 Quiz 2- Requires Respondus LockDown Browser

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