Question

The Melodic Kortholt Company will change its current health plan if at least half the employees are dissatisfied with it. A trial sample of 25 employees shows that 16 are dissatisfied. The p-value for a right-tailed test is a. 0.4192 b. 0.1337 c. 0.0808 d. 0.0901

Answer 1

Concepts and reason

Statistical hypotheses testing: Hypotheses testing is used to make inferences about the population based on the sample data. The hypotheses test consists of null hypothesis and alternative hypothesis.

Null hypothesis: The null hypothesis states that there is no difference in the test, which is denoted by ${H_0}$ . Moreover, the sign of null hypothesis is equal $\left( = \right)$ , greater than or equal $\left( \ge \right)$ and less than or equal $\left( \le \right)$ .

Alternative hypothesis: The hypothesis that differs from ${H_0}$ is called alternative hypothesis. This signifies that there is a significant difference in the test. The sign of alternative hypothesis is less than $\left( < \right)$ , greater than $\left( > \right)$ , or not equal $\left( \ne \right)$ .

Proportion: The ratio of the number of favorable outcomes and total number of possible outcomes in the sample called as the proportion.

Z-statistic for proportion: The standardized z- statistic is defined as the ratio of the ‘distance of the observed statistic from the proportion of the null distribution’ and the ‘standard deviation of the null distribution’.

P-value: The probability of getting the value of the statistic that is as extreme as the observed statistic when the null hypothesis is true is called as P-value. Therefore, it assumes “null hypothesis is true”.

Fundamentals

The formula for sample proportion is,

${\rm{Sample proportion}}\left( {\hat p} \right) = \frac{x}{n}$

Where x is the number of success in the sample and n is the total sample size.

The formula for population proportion is,

${\rm{Population proportion}}\left( {{p_0}} \right) = \frac{X}{N}$

Where X is the number of success in the population and N is the total population size.

The test statistic for one proportion is,

$z = \frac{{\hat p - {p_0}}}{{\sqrt {\frac{{{p_0}\left( {1 - {p_0}} \right)}}{n}} }}$

Here,

$\begin{array}{c}\\\hat p:{\rm{Sample}}\,{\rm{proportion}}\\\\{p_0}:{\rm{Population}}\,{\rm{proportion}}\\\\n:{\rm{Sample}}\,{\rm{size}}\\\end{array}$

The formula for the p-value for right-sided test is, $p{\rm{ - value}} = P\left( {Z > z} \right)$

Procedure for finding the probability is listed below:

1.From the table of standard normal distribution, locate the z-value

2.Move left until the first column is reached.

3.Move upward until the top row is reached.

4.Locate the probability value, by the intersection of the row and column values gives the area to the left of z.

From information given, out of 25 employees, 16 are dissatisfied.

$\begin{array}{c}\\\hat p = \frac{x}{n}\\\\ = \frac{{16}}{{25}}\\\\ = 0.64\\\end{array}$

The value of z-statistic is obtained as shown below:

$\begin{array}{c}\\z = \frac{{\hat p - {p_0}}}{{\sqrt {\frac{{{p_0}\left( {1 - {p_0}} \right)}}{n}} }}\\\\ = \frac{{0.64 - 0.5}}{{\sqrt {\frac{{0.5\left( {1 - 0.5} \right)}}{{25}}} }}\\\\ = \frac{{0.14}}{{0.10}}\\\\ = 1.4\\\end{array}$

The P-value is obtained as shown below:

The value of z statistic is 1.4.

$\begin{array}{c}\\P - {\rm{value}} = P\left( {Z > z} \right)\\\\ = P\left( {Z > 1.4} \right)\\\\ = 1 - P\left( {Z \le 1.4} \right)\\\end{array}$

Procedure for finding the probability is listed below:

1.From the table of standard normal distribution, locate the z-value as $z = 1.4$

2.Move left until the first column is reached. Note the value as 1.4

3.Move upward until the top row is reached. Note the value as 0.00.

4.Locate the probability value as 0.9192 by the intersection of the row and column values gives the area to the left of $z = 1.4$ .

$\begin{array}{c}\\P{\rm{ - value}} = 1 - 0.9291\\\\ = 0.0808\\\end{array}$

Ans:

The P-value for a right tailed test is 0.0808.