Chapter 15, Problem 1CQ

a.

To determine

To find:The value of test statistic.

Answer to Problem 1CQ

The value of test statistic is $-2.7$ .

Explanation of Solution

Given information:

The data is $50,38,23,29,57,26,53,42,60,23$ and the hypothesis are $H_0\text{:}m=30$ and $H_1\text{:}m\ne 30$ .

Concept used:

The test statistic is,

If the sample size is less than or equal to $25$ then the test statistic is $X$ .

If the sample size is greater than $25$ then the test statics is,

  $z=\frac{x+0.5-\frac{n}{2}}{\frac{\sqrt{n}}{2}}$

Here, the number of times the less frequent sign occur is $x$ .

The conditions for test statistics are shown below in table.

Type of test	Test statistic
Right-tailed test	Number of minus signs
Left-tailed test	Number of plus signs
Two-tailed test	Number of plus or minus signs, whichever is smaller

Calculations:

The value of test statistic is shown below.

Sample	Difference	Signs
50	20	-
38	8	+
23	-7	-
29	-1	-
57	27	+
26	-4	-
53	23	+
42	12	+
60	30	+
23	-7	-

From the above table, the number of minus sign is $4$ and the number of plus sign is $6$ and the sample size is $19$ .

Since, the sample size is less than $25$ .

The test statistic is given by the number of plus or minus sign which are smaller in number.

  $x=4$

Therefore, the value of test statistic for the given sample is $2$ .

b.

To determine

To find:Whether the hypothesis $H_0$ rejected or not.

Answer to Problem 1CQ

The hypothesis $H_0$ is not rejected.

Explanation of Solution

Given information:

The level of significance is $0.05$ .

Calculations:

The critical value for two-tailed test at sample size of $10$ and level of significance of $0.05$ is $1$ .

Since, the test statistics is $4$ which greater than the critical value.

Therefore, the hypothesis $H_0$ is not rejected.

c.

To determine

To find:Whether the hypothesis $H_0$ rejected or not.

Answer to Problem 1CQ

The hypothesis $H_0$ is not rejected.

Explanation of Solution

Given information:

The level of significance is $0.01$ .

Calculations:

The critical value for two-tailed test at sample size of $10$ and level of significance of $0.01$ is $0$ .

Since, the test statistics is $4$ which greater than the critical value.

Therefore, the hypothesis $H_0$ is not rejected.