Question

A report shows that the average amount of money spent on public transportation (taxis, subway, etc.) by millennials in Philadelphia, PA (per week) is about $78, with a standard deviation of $19. The exact distribution is not known, but a random sample of 56 millennials from Philadelphia is selected. a. Describe the sampling distribution of the sample mean x (4 pts) The mean: x = _________________ The standard error on the mean (round to 2 dec. places):  x = ________________ The type/shape of the sampling distribution is ____________________________ Why are you allowed to make this claim about the type of distribution? b. Determine the probability that the average amount spent weekly on public transportation by the 56 milleniials in this study would be between $75 and $85. (Be sure to include an accurate sketch!)

Answer 1

a)

mena = 78 ,

std.deviation = 19/sqrt(56) = 2.54

The shapr of distribution is normal

b)

Here, μ = 78, σ = 2.54, x1 = 75 and x2 = 85. We need to compute P(75<= X <= 85). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (75 - 78)/2.54 = -1.18
z2 = (85 - 78)/2.54 = 2.76

Therefore, we get
P(75 <= X <= 85) = P((85 - 78)/2.54) <= z <= (85 - 78)/2.54)
= P(-1.18 <= z <= 2.76) = P(z <= 2.76) - P(z <= -1.18)
= 0.9971 - 0.119
= 0.8781