Respuesta :
Answer:
[tex]\hat p =0.5 -1.28 \sqrt{\frac{0.5}{100}}=0.4359[/tex]
Step-by-step explanation:
1) Data given and notation
n=100 represent the random sample taken
X=40 represent the female new Ph.D.’s in economics
[tex]\hat p=\frac{40}{100}=0.4[/tex] estimated proportion of female new Ph.D.’s in economics
[tex]p_o=0.5[/tex] is the value that we want to test
[tex]\alpha=0.1[/tex] represent the significance level
Confidence=90% or 0.90
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim given by: the null hypothesis that the proportion of new Ph.D.’s in economics who are female is at least 0.5, against the alternative hypothesis that it is les.:
Null hypothesis:[tex]p\geq 0.5[/tex]
Alternative hypothesis:[tex]p < 0.5[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
3) Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.4 -0.5}{\sqrt{\frac{0.5(1-0.5)}{100}}}=-2[/tex]
4) Critical value
For this case we can find the critical value with the normal standard distribution we need a value that accumulates 0.1 of the area on the left tail. And we can find it with the following excel code :"=NORM.INV(0.1;0;1)". And the critical value owuld be [tex]z_{\alpha/2}=-1.28[/tex].
Now in order to find the critical value in terms of p we know this:
[tex]z = \frac{\hat p -p_0}{\sqrt{\frac{p_0 (1-p_0)}{n}}}[/tex]
and replacing the values that we got :
[tex]-1.28 = \frac{\hat p -0.5}{\sqrt{\frac{0.5 (1-0.5)}{100}}}[/tex]
And solving for [tex]\hat p[/tex] we got:
[tex]\hat p =0.5 -1.28 \sqrt{\frac{0.5}{100}}=0.4359[/tex]
