The probability that the sample mean is in between 50 minutes and 51 minutes is 0.345.
n = sample size = 39
σ = standard deviation =4.2
μ = population mean = 49.8
Probability is the chance that an event can occur, in this case probability is used to measure the chance sample mean is between 50 minutes and 51 minutes. So,
P(50 ≤ x ≤ 51) = P([tex]\frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex] ≤ z ≤ [tex]\frac{\bar x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex])
= P([tex]\frac{50-49.8}{\frac{4.2}{\sqrt{39}}}[/tex] ≤ z ≤ [tex]\frac{51-49.8}{\frac{4.2}{\sqrt{39}}}[/tex])
= P(0.30 ≤ z ≤ 1.78)
= P(z ≤ 1.78) - P(z ≤ 0.3)
using z table and we get,
= 0.9625 - 0.6179
= 0.3446
= 0.345
Thus, the probability is 0.345 for the sample mean is within 50 and 51.
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