Question

)The thaxel tbeuen an arpat and a cents Giłis sampad for Public bises and Pmieke tasvat's. A sampe of cigkt bs tups yel to 4,5, 31,42 Ton fou (abs re included in tk Survey uwith times or 37, ч6/ 40, 327 35/46/33 41 24,3C n iven te trax) tedffcne, can we concide thet te tax cabs get te ndes to ityaa Sgnifintly fastr tire

Answer 1

Data:

	Private Taxi ($\mu _{1}$	)Public Buses ( μ2 )
	37	44
	46	56
	40	39
	32	42
	35	48
	46	51
	39	36
	41	43
	29
	36
Sample Mean	38.1	44.875
Sample Std. Deviation	5.506561742	6.5123509

A.

We know that,

Confidence Interval on a Difference in Means: ...ncs known, small samples Populations are normally distributed The population variances are assumed not equal (σ14 σ22). the test statistic t' " - m, n is approximately distributed as t, with degrees of freedom given by si l'ーー-(,,2 called Smith-Satterthwaite degrees of freedom In) (5 /n) 1,-1 n2-1 Ifv is not integer then round down to an integer

and

Confidence Interval on a Difference in Means: Variances unknown, small samples Populations are normally distributed The population variances are assumed equal (σ2-022-02), so use the two sample variances and pool them to estimate σ2 the test statistic is a t value with (m + n2-2) degrees of freedom. - The pooled estimator of ơ, denoted by s, is defined by (n2-1)s 2 i1 n +n2-2 DIITKGP

H_{0 : $\mu _{1} - \mu _{2} < 0$}

H₁ : $\mu _{1} - \mu _{2} \geq 0$

Observed difference (Sample 1 - Sample 2): -6.775

If populations are assumed to have unequal variances then,

Standard Error of the Difference (S_e)

cool .

=

5.506566.5123509 10

=

30.3222030336 42.4107142447 1 10

=

8.3336 8.586

=

2.8868

Standard Deviation of Difference: 2.8868

DF: 13

Test Statistic t = -2.3469

P-Value =0.9823 > 0.05 i.e H₀ can't be rejected and hence we can say that statistically private taxis are faster than (takes less time) public buses

If populations are assumed to have equal variances then,

Equal Variances
Pooled Standard Deviation: 5.9675
Pooled DF: 16

Test Statistic t= -2.3935
Population 1 < Population 2: P-Value = 0.9854   > 0.05 i.e H₀ can't be rejected and hence we can say that statistically private taxis are faster than (takes less time) public buses.

Thus in both cases, the result is the same.

B.

The following null and alternative hypotheses need to be tested:

Ho: σ₁² = σ₂²

Ha: σ₁²$\neq$ σ₂²

The F-statistic is computed as follows:

F = 42.41071429/30.32222222 =0.715

Rejection region:

R={F: F< 0.238 or F>4.823}.

Since F statistics doesn't fall in the rejection, it is then concluded that the null hypothesis is not rejected. σ₂²

Therefore, there is not enough evidence to claim that the population σ₁² is different than the population σ₂² at the α=0.05 significance level.

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