Normal distribution Test

{"questions":[{"content":"For X ~ N(72, 8²), the z-value at x = 88 is [[input-nor-q1]]","widgets":{"input-nor-q1":{"type":"input","answer":"2"}},"hints":["Use (x − μ)/σ."],"explanation":"[M1] (88 − 72)/8. [A1] z = 2.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["M1","A1"],"follow_through":false}},{"content":"For X ~ N(30, 4²), which setup finds P(X > 37)?<br>[[choice-nor-q2]]","widgets":{"choice-nor-q2":{"type":"choice","options":["Normal CDF from 37 to an upper infinity bound","Inverse normal with area 37","Normal CDF from 30 to 37 only","Regression of 30 on 4"],"answer":[0],"explanations":["[M1] Identify the lower bound 37. [A1] Use μ = 30 and σ = 4 in normal CDF.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["This is a right-tail area."],"explanation":"[M1] Identify the lower bound 37. [A1] Use μ = 30 and σ = 4 in normal CDF.","assessment":{"paper_mode":"paper_2","calculator_mode":"expected","gdc_skill":"normal","marks":["M1","A1"],"follow_through":false}},{"content":"The upper 5% boundary is required. Which cumulative area should inverse normal use?<br>[[choice-nor-q3]]","widgets":{"choice-nor-q3":{"type":"choice","options":["0.95","0.05","1.05","−0.95"],"answer":[0],"explanations":["[M1] Convert the upper tail. [A1] Left-tail area = 0.95.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Inverse normal commonly takes area to the left."],"explanation":"[M1] Convert the upper tail. [A1] Left-tail area = 0.95.","assessment":{"paper_mode":"paper_2","calculator_mode":"expected","gdc_skill":"inverse_normal","marks":["M1","A1"],"follow_through":false}},{"content":"A symmetric interval is μ − 2σ < X < μ + 2σ. Which statement is true?<br>[[choice-nor-q4]]","widgets":{"choice-nor-q4":{"type":"choice","options":["The two excluded tail probabilities are equal","All probability lies inside","The mean lies outside","The variance must be 2"],"answer":[0],"explanations":["[R1] Equal distances from μ give equal tail areas.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Use symmetry about μ."],"explanation":"[R1] Equal distances from μ give equal tail areas.","assessment":{"paper_mode":"paper_1","calculator_mode":"none","gdc_skill":"","marks":["R1"],"follow_through":false}},{"content":"A student enters variance instead of standard deviation in normal CDF. What is the likely effect?<br>[[choice-nor-q5]]","widgets":{"choice-nor-q5":{"type":"choice","options":["The spread used by the calculator is wrong","The mean automatically changes","The probability is always 0.5","Nothing changes"],"answer":[0],"explanations":["[R1] The calculator expects σ; entering σ² changes the modeled spread and probability.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Check what the calculator parameter represents."],"explanation":"[R1] The calculator expects σ; entering σ² changes the modeled spread and probability.","assessment":{"paper_mode":"paper_2","calculator_mode":"expected","gdc_skill":"normal","marks":["R1"],"follow_through":false}}],"mix":1}