For the following hypothesis test:

With n = 80, σ = 9, and = 47.1, state the calculated value of the test statistic z.

Question

For the following hypothesis test:

With n = 80, σ = 9, and  = 47.1, state the calculated value of the test statistic z.

BlackTom AI · Accepted Answer

To start, we identify the key values provided: n = 80, σ = 9, and x̄ = 47.1, with the null hypothesis μ ≤ 45 and alternative μ > 45. The test statistic for a z-test with known population standard deviation is computed as z = (x̄ − μ0) / (σ / √n). Option by option analysis: Option A: 3.151 If z were 3.151, that would imply a very large difference between x̄ and μ0 relative to the standard error. Specifically, z ≈ 3.15 would require (x̄ − μ0) ≈ 3.15 × (σ / √n). With σ/√n ≈ 9 / √80 ≈ 1.006, the numerator would need to be ≈ 3.15, but the observed difference is x̄ − μ0 = 47.1 − 45 = 2.1. This makes z around 2.1, not 3.15, so this option is inconsistent with the data. Option B: -3.121 A negative z-value would indicate x̄ < μ0, but here x̄ = 47.1 is greater than μ0 = 45, yielding a positive numerator (2.1). Therefore a negative z-value is incompatible with the sign of the observed difference, and this option is incorrect. Option C: -2.141 Similar to the previous negative options, this would require x̄ < μ0 to produce a negative z, which contradicts x̄ > μ0. Since the observed difference is positive, this option cannot be correct. Option D: 2.087 With the numbers plugged in: σ/√n ≈ 9 / √80 ≈ 9 / 8.944 ≈ 1.006, and x̄ − μ0 = 2.1. Then z = 2.1 / 1.006 ≈ 2.087, which matches the observed difference and the given parameters. This option aligns with the standard z-test calculation. Overall, the calculations show that the only option consistent with x̄ = 47.1, μ0 = 45, and σ = 9 with n = 80 is a positive z around 2.087, while the other options either have the wrong sign or magnitude given the same data.

For the following hypothesis test: With n = 80, σ = 9, and = 47.1, state the calculated value of the test statistic z. 单项选择题

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