Question

MANUF	POP
213	582
91	132
453	716
454	515
412	158
80	80
434	757
136	529
207	335
368	497
3344	3369
361	746
104	201
125	277
291	593
204	361
625	905
1064	1513
699	744
381	507
775	622
181	347
46	244
44	116
391	463
462	453
1007	751
266	540
1692	1950
347	520
343	179
337	624
275	448
641	844
721	1233
137	176
96	308
197	299
379	531
35	71
569	717

It is commonly believed that cities with population of 500 and higher have on average higher manufacturing than the cities with population of less than 500. Use Pollution data and your statistical expertise to answer the questions: Is this a reasonable belief?

4. What test/procedure did you perform?

• a. One-sided t-test
• b. Two-sided t-test
• c. Regression
• d. ​​Confidence interval

5. Statistical Interpretation

• a. Since P-value is small we are confident that the slope is not zero.
• b. Since P-value is small we are confident that the averages are different.
• c. Since P-value is too large the test is inconclusive.
• d. ​​None of these.

6. Conclusion

• a. Yes, I am confident that the above belief is correct.
• b. No, I cannot claim that the above belief is correct.

Task 1

Answer 1

Solution:

Here, we have to use two sample t test for population means assuming equal population variances.

Null hypothesis: H₀: Cities with population of 500 and higher have same manufacturing as the cities with population of less than 500.

Alternative hypothesis: H_a: Cities with population of 500 and higher have on average higher manufacturing than the cities with population of less than 500.

H₀: µ₁ = µ₂ versus H_a: µ₁ > µ₂

This is an upper tailed (one tailed) test.

µ₁ = Average manufacturing of cities with population with 500 or more.

µ₂ = Average manufacturing of cities with population less than 500

Test statistic formula for pooled variance t test is given as below:

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

We are given

X1bar = 690.4090909

X2bar = 199.8947368

S1 = 686.992733

S2 = 136.623121

n1 = 22

n2 = 19

df = n1 + n2 – 2 = 22 + 19 – 2 = 39

α = 0.05

Critical t value = 1.6849

(by using t-table)

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(22 – 1)* 686.992733^2 + (19 – 1)* 136.623121^2]/(22 + 19 – 2)

S_p² = 262746.7976

t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

t = (690.4090909 – 199.8947368) / sqrt[262746.7976*((1/22)+(1/19))]

t = 490.5144 / 160.5360

t = 3.0555

P-value = 0.0020

(by using t-table)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that Cities with population of 500 and higher have on average higher manufacturing than the cities with population of less than 500.

4. What test/procedure did you perform?

Answer: a. One-sided t test

5. Statistical Interpretation

Answer: d. ​​None of these.

6. Conclusion

Answer: a. Yes, I am confident that the above belief is correct.