Question

A 0.05 significance level is used for a hypothesis test of the claim that when parents use a particular method of gender​selection, the proportion of baby girls is less than 0.5. Assume that sample data consists of 66 girls in 144 ​births, so the sample statistic of 11/ 24 EndFraction 11 24 results in a z score that is 1 standard deviation below 0. a) what is the pvalue b) critic value c) area of the critical region

Answer 1

a)
`z=\frac{0.458 -0.5}{\sqrt{(0.5(1-0.5))/(144)}}=-1.008`

Since is a left tailed test the p value would be:

`p_v =P(z<-1.008)=0.1567`

b)
`z_(crit)= -1.645`

c) For this case since is a left tailed test the critical region or the rejection zone of the null hypothesis would be:

`(\infty , -1.645)`

Explanation:

Data given and notation

n=144 represent the random sample taken

X=66 represent the number of girls

`\hat p=(66)/(144)=0.458` estimated proportion of girls

`p_o=0.5` is the value that we want to test

`\alpha=0.05` represent the significance level

Confidence=95% or 0.95

z would represent the statistic (variable of interest)

`p_v` represent the p value (variable of interest)

Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that true proportion is less than 0.5.:

Null hypothesis:
`p\geq 0.5`

Alternative hypothesis:
`p < 0.5`

When we conduct a proportion test we need to use the z statistic, and the is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

The One-Sample Proportion Test is used to assess whether a population proportion
`\hat p` is significantly different from a hypothesized value
`p_o`.

Calculate the statistic

Since we have all the info requires we can replace in formula (1) like this:

`z=\frac{0.458 -0.5}{\sqrt{(0.5(1-0.5))/(144)}}=-1.008`

Part a : p value

The significance level provided
`\alpha=0.05`. The next step would be calculate the p value for this test.

Since is a left tailed test the p value would be:

`p_v =P(z<-1.008)=0.1567`

Part b

We want to conduct a left tailed test with
`\alpha=0.05` and we need to find a critical value in the normal standard distribution who accumulates 0.05 of the area in the left and we got:

`z_(crit)= -1.645`

Part c

For this case since is a left tailed test the critical region or the rejection zone of the null hypothesis would be:

`(\infty , -1.645)`

Answer 2

a) P-value=0.16

b) critic value zc=-1.645

c) area of the critical region = 0.05

Explanation:

We have a hypothesis test on the population proportion.

The sample proportion is p=66/144=11/24=0.4583.

The z-statistic results in z=-1.

As the alternative hypothesis (the claim) states that the proportion is below 0.5, the test is left-tailed.

The P-value for this type of test and with a z=-1 is calculated as:

`P-value=P(z<-1)=0.16`

The critial value depends on the significance level. In this test we have a significance level of 0.05.

The critical value can be looked-up in a standard normal distribution table. Its the value for which the probability of having a statistic below this critical value is equal to the significance level (0.05):

`P(z<z_c)=0.05`

For this test, the critical value is zc=-1.645.

As the área of the critical region is equal to the probability of having a statistic below this critical value, and this is the significance level, we have a área of the critical region equal to 0.05.