MGMT2263 Tutorial Sheet 1 Solutions
1) Ho: m ³ 125
Ha: m < 125
Reject Ho if p-value < 1%. Do not reject Ho if p-value > 10%.
Z = (110.03 – 125)/20.5/sqrt(12) = -2.53
p-value = P(Z < -2.53) = 0.5 – 0.4943 = 0.0057
Since the p-value < 1%, reject Ho
Conclude that customers spend less than $125 on average.
2) Reject Ho if Z < -1.645
Reject Ho if (xbar – 125)/(20.5/sqrt(12)) < -1.645
Reject Ho if xbar < 125 – 1.645(20.5)/sqrt(12)
Reject Ho if xbar < 115.2652
Power = P(reject Ho | Ha true)
Power = P(xbar < 115.2652 | m = 110)
Power = P[Z < (115.2652 – 110)/(20.5/sqrt(12))]
Power = P(Z < 0.89) = 0.5 + 0.3133 = 0.8133 = 81.33%
3) We know that we reject Ho if xbar < 115.2652.
In order for the power to be 99%, the Z value must be 2.326 since P(Z < 2.326) = 0.99.
Then, (115.2652 - m)/(20.5/sqrt(12)) = 2.326
m = 115.2652 – 2.326(20.5)/sqrt(12) = 101.5 after rounding. The alternative mean would need to be $101.50
4) Ho: m = 0
Ha: m ¹ 0
Reject Ho if Z < -1.96 or Z > 1.96
Z = (0.02 – 0)/0.0832/sqrt(40) = 1.52
Do not reject Ho
Conclude there is no significant difference between this clock system and GMT.
5) Lower limit = 0.02 – 1.96(0.0832)/sqrt(40) = 0.02 – 0.0258 = -0.0058
Upper limit = 0.02 + 0.0258 = 0.0458
Since the hypothesized mean of 0 falls inside the 95% confidence interval, we do not reject Ho at a 5% level of significance.
6) p-value = 2P(Z > 1.52) = 2(0.0643) = 0.1286. Since the p-value is greater than 10%, there is no level of significance between 1% and 10% that could have been chosen in which the opposite conclusion would have been reached.
7) Accept Ho if –1.96 < Z < 1.96
Accept Ho if –1.96 < (xbar – 0)/(0.0832/sqrt(40)) < 1.96
Accept Ho if –1.96(0.0832)/sqrt(40) < xbar < 1.96(0.0832)/sqrt(40)
Accept Ho if –0.0258 < xbar < 0.0258
b = P(Accept Ho | Ha true)
b = P(–0.0258 < xbar < 0.0258 | m = 0.02)
b = P[(-0.0258 – 0.02)/(0.0832/sqrt(40)) < Z < (0.0258 – 0.02)/(0.0832/sqrt(40))]
b = P(-3.48 < Z < 0.44) = 0.4998 + 0.17 = 0.6698 = 66.98%
8) Ho: median £ 7
Ha: median > 7
There are two values of 7 in the data. Therefore, we reduce the sample size to 8 from 10. We reject Ho if T- £ 6.
To use Megastat, put the 8 values of the data in column A and the hypothesis median of 7 in each of the corresponding cells in column B from B1 through B8. The Megastat output is:
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Wilcoxon Signed
Rank Test |
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variables: |
Group
1 - Group 2 |
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27.5 |
sum
of positive ranks |
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8.5 |
sum
of negative ranks |
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8 |
n |
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18.00 |
expected value |
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7.14 |
standard deviation |
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1.33 |
z |
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.0917 |
p-value (one-tailed, upper) |
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We can ignore the Z stuff in the last 2 lines since the sample size is not large enough. We have T- = 8.5. Do not reject Ho. The median rating is not significantly higher than 7. Therefore, the manufacturer should not proceed with this peanut butter.
9) Ho: m ³ 20
Ha: m < 20
Since the sample size is 8, the degrees of freedom is 7. Reject Ho if p-value < 1%. Do not reject Ho if p-value > 10%.
t = (18.5 – 20)/6.3246/sqrt(8) = -0.6708
p-value = P(t < -0.6708). From the t table, we see that the closest critical value for 7 degrees of freedom is the 10% critical value of 1.415. This means P(t < -1.415) = 10%. Since our test statistic is to the right of –1.415, this means P(t < -0.6708) > 10%. Therefore we do not reject Ho.
We conclude people do not watch less than 20 hours of TV a week on average.
If we use Megastat for the analysis, here is the output:
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Hypothesis
Test: Mean vs. Hypothesized Value |
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20.000 |
hypothesized
value |
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18.500 |
mean
Data |
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6.325 |
std.
dev. |
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2.236 |
std.
error |
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8
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n |
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7
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df |
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-0.6708 |
t |
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.2619 |
p-value (one-tailed, lower) |
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From the output, we see the exact p-value of 0.2619, confirming that the p-value is greater than 10%.
10) Ho: s = 10
Ha: s ¹ 10
As with the t test in question 9, the degrees of freedom is 7. Reject Ho if c2 <1.6899 or >16.0128.
c2 = 7(6.3246)2/(10)2 = 2.8
Do not reject Ho
Conclude there is no significant difference in the standard deviations of the group and the general population.
If we use Megastat for the analysis, here is the output:
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Chi-square
Variance Test |
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100.000 |
hypothesized
variance |
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40.000 |
observed
variance of Data |
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8
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n |
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7
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df |
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2.80 |
chi-square |
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.1943 |
p-value (two-tailed) |
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17.486 |
confidence
interval 95.% lower |
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165.694 |
confidence
interval 95.% upper |
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11) Lower limit = sqrt[7(6.3246)2/16.0128] = 4.1816
Upper limit = sqrt[7(6.3246)2/1.6899] = 12.8722
Since the hypothesized standard deviation of 10 falls inside the 95% confidence interval, we do not reject Ho at a 5% level of significance.
Note that if you were to use the Megastat output, you would need to square root the values of 17.486 and 165.694 since these are the confidence interval limits for the population variance.
12) Since 6.3246 < 10, p-value = 2P(c2 < 2.8). From the chi-square table with 7 degrees of freedom, we see that 2.8 is between the 95% critical value of 2.1683 and the 90% critical value of 2.8331. This means P(c2 > 2.1683) = 0.95 and P(c2 > 2.8331) = 0.9. Conversely, this means P(c2 < 2.1683) = 0.05 and P(c2 < 2.8331) = 0.1. So, P(c2 < 2.8) is between 5% and 10% (although closer to 10% than 5%). By extension, the p-value = 2P(c2 < 2.8) is between 10% and 20%. Since the p-value is greater than 10%, we would not reject Ho under the general rule of thumb.
From the Megastat output, we see the exact p-value of 0.1943, confirming that the p-value is greater than 10%.
13) Ho: p £ 50%
Ha: p > 50%
Reject Ho if Z > 1.645
p-hat = 220/400 = 0.55
Z = (0.55 – 0.5)/sqrt(0.5*0.5/400) = 2
Reject Ho
Conclude the percentage of household with an income above $50,000 exceeds 50%.
14) Reject Ho if Z > 1.645
Reject Ho if (phat – 0.5)/sqrt(0.5(0.5)/400) > 1.645
Reject Ho if phat > 0.5 + 1.645sqrt(0.5(0.5)/400)
Reject Ho if phat > 0.5411
Power = P(reject Ho | Ha true)
Power = P(phat > 0.5411 | p = 0.55)
Power = P(Z > (0.5411 – 0.55)/sqrt(0.55(0.45)/400))
Power = P(Z > -0.36) = 0.5 + 0.1406 = 0.6406 = 64.06%
15) Lower limit = 0.55 – 1.96sqrt(0.55(0.45)/400) = 0.55 – 0.049 = 50.1%
Upper limit = 0.55 + 0.049 = 59.9%
If we use Megastat for the analysis, here is the output:
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Hypothesis test
for proportion vs hypothesized value |
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Observed |
Hypothesized |
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0.55 |
0.5 |
p (as decimal) |
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220/400 |
200/400 |
p (as fraction) |
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220. |
200. |
X |
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400 |
400 |
n |
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0.025 |
std. error |
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2.00 |
z |
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.0228 |
p-value (one-tailed, upper) |
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0.5012 |
confidence
interval 95.% lower |
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0.5988 |
confidence
interval 95.% upper |
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0.0488 |
margin of error |
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