STAT217 midterm solutions

Question 1

a) Ho: p1 £ p2

Ha: p1 > p2

Reject Ho if Z > 1.645

phat = 400/650 = 0.6154

Z = (0.64 – 0.6)/sqrt(0.6154(0.3846)/400 + 0.6154(0.3846)/400) = 1.02

Do not reject Ho.

Conclude the percentage of households with incomes above $100,000 that recycle is not significantly higher than households with incomes under $35,000.

b) P-value = P(Z > 1.02) = 0.5 – 0.3461 = 0.1539. We reach the same conclusion since the p-value > 10%.

Question 2

a) Reject Ho if (xbar – 150)/(78.27/sqrt(40)) > 1.645

Reject Ho if xbar > 150 + 1.645(78.27)/sqrt(40)

Reject Ho if xbar > 170.3578

Power = P(xbar > 170.3578 | m = 175)

Power = P(Z > (170.3578 – 175)/(78.27/sqrt(40)) = P(Z > -0.38) = 0.5 + 0.148 = 0.648 = 64.8%

b) –1.282 = (170.3578 - m)/(78.27/sqrt(40)) from which we get m = 186.22 after rounding to the nearest cent.

c) Any of: increase n, increase alpha or large alternative mean.

Question 3

a) Ho: m1 £ m2

Ha: m1 > m2

Degrees of freedom = 12; Reject Ho if t > 1.782

S_p² = (5.6667 + 8)/2 = 6.83333

t = (13 – 10)/sqrt(6.83333/7 + 6.83333/7) = 2.147

Reject Ho.

Conclude the older group spends significantly more time on average reading newspapers.

b) Between 2.65% and 10%.

c) Lower limit = (13 – 10) – 2.179 sqrt(6.83333/7 + 6.83333/7) = 3 – 3.0447 = 0

Upper limit = 3 + 3.0447 = 6

0 < m1 - m2 < 6

Question 4

Ho: d £ 0

Ha: d > 0

Reject Ho if T- £ 4 (based on d = after – before)

T- = 1

Reject Ho.

Conclude the patrons like the new décor better.