Question

Suppose we have a set of paired data, and let d = xafter − xbefore . The null hypothesis is H0 : μd = 0, the alternative hypothesis is Ha : μd ≠ 0, and the population standard deviation σ is not known. Jordan collects a sample of size n = 9 and computes d¯= 2.2 and sd = 3.3A) What is the value of the test statistic?B) The probability of obtaining a test statistic greater than Jordan's test statistic is 0.0403. Use this probability to determine the P-value for Jordan's hypothesis test. (Round to three decimal places.)

Answer 1

Answer:

(a) The value of the test statistic is, t = 2.

(b) The p-value of the test is 0.081.

Step-by-step explanation:

The hypothesis for the paired difference test is:

H₀: $\mu_{d}=0$ vs. Hₐ: $\mu_{d}\neq 0$

Given:

$\bar d=2.2\\SD_{d}=3.3\\n=9$

(a)

The test statistic is:

$t=\frac{\bar d}{\frac{SD_{d}}{\sqrt{n}} } \\=\frac{2.2}{\frac{3.3}{\sqrt{9}} }\\=2$

The value of the test statistic is, t = 2.

(b)

It is provided that the $P(T>2)=0.0403$.

But the hypothesis test is two-tailed.

The p-value for the two-tailed test is:

$p-value=2P(t>2)$ or $p-value=2P(t<2)$

The p-value is:

$p-value=2P(t>2)=2\times0.0403=0.0806\approx0.081$

Thus, the p-value of the test is 0.081.

Answer 2

Using the paired t - test statistic, the value of the test statistic in the scenario above is 2 with a p-value of 0.081

Using the test statistic relation for a paired sample :

$t = \frac{d}{\frac{S_{d}}{\sqrt{n}}}$

• d = 2.2

• $S_{d} = 3.3$

• n = 9

Plugging the values into the test statistic equation :

$t = \frac{2.2}{\frac{3.3}{\sqrt{9}}} = \frac{2.2}{1.1}= 2$

Hence, the test statistic = 2

Probability of obtaining a test statistic value greater than 2 = 0.0403

• P(t > 2) = 0.0403

Since, it is a 2 - tailed test ;

P(t > 2) = 0.0408 × 2 = 0.0806 = 0.081 (3 decimal places).

Therefore, the p value for the hypothesis is 0.081.

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