Question

The paint used to make lines on roads must reflect enough light to be clearly visible at night. Let ? denote the true average reflectometer reading for a new type of paint under consideration. A test of H₀: ? = 20 versus H_a: ? > 20 will be based on a random sample of size n from a normal population distribution. What conclusion is appropriate in each of the following situations? (Round your P-values to three decimal places.)

(a)    n = 17, t = 3.1, ? = 0.05
P-value =
(b)    n = 10, t = 1.7, ? = 0.01

P-value =  
(c)    n = 29,

t = ?0.3 P-value = ?

Answer 1

Concepts and reason

Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data.

Hypothesis testing is basically an assumption that we make about the population parameter. It is introduced by the Ronald Fisher.

Hypothesis testing for the testing of difference between two sample means is used to test the difference between two independent population means.

Hypothesis is a statement or assertion about the unknown population parameter.

Fundamentals

The procedure is as follows:

Step-1: Make the Null hypothesis.

Step-2: Make Alternative hypothesis.

Step-3: Identify the level of significance and decision rule using critical value or p-value.

Step-4: Calculate test statistic.

Step-5: Comparison and making decision about the Null hypothesis.

The formula for calculating the test statistic for t of single mean is,

The formula of degrees of freedom is,

The Excel’s formula for calculating p-value is,

a)

The Null and Alternative hypothesis are as follows:

Null hypothesis,

Alternative hypothesis,

The sample information is as follows:

The degrees of freedom are,

The calculation of the p-value is,

From the p-value and level of significance, observe that the null hypothesis is rejected.

Hence, conclude that the population mean is greater than 20.

b)

The Null and Alternative hypothesis are as follows:

Null hypothesis,

Alternative hypothesis,

The sample information is as follows:

The degrees of freedom are,

The calculation of the p-value is,

From the p-value and level of significance, observe that the null hypothesis is fails to reject.

Hence, conclude that the population mean is not greater than 20.

c)

The Null and Alternative hypothesis are as follows:

Null hypothesis,

Alternative hypothesis,

The sample information is as follows:

Assume that the level of significance is 0.05

The degrees of freedom are,

The calculation of the p-value is,

From the p-value and level of significance, observe that the null hypothesis is fails to reject.

Hence, conclude that the population mean is not greater than 20.

Ans: Part a

The Null hypothesis is rejected.

The population mean is greater than 20.

Part b

The Null hypothesis is fails to reject.

The population mean is not greater than 20.

Part c

The Null hypothesis is fails to reject.

The population mean is not greater than 20.