Question 1201344
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Part (a)


Null: p = 0.3
Alternative: p > 0.3


Rule: The null ALWAYS will have the equal sign. 
This is to lock down the parameter we're trying to test.
The alternative will have an inequality sign of some sort (eg: "greater than")


The claim is p > 0.3 which tells us the direction of the test. We're doing a right-tailed test. 
The rejection region is to the right of the critical value because of this right-tailed test.


At the 5% level of significance, the z critical value is roughly z = 1.645
This is because P(Z > 1.645) = 0.05
Approximately 5% of the area under the Z curve is to the right of 1.645.
Use a table or calculator to determine this value.


Use a Z table such as this
<a href = "https://www.ztable.net/">https://www.ztable.net/</a>
or the table found in the back of your stats textbook


Use such a table to find that
P(Z < 1.00) = 0.84134
so,
P(Z > 1.00) = 1 - P(Z < 1.00)
P(Z > 1.00) = 1 - 0.84134
P(Z > 1.00) = 0.15866
This is the approximate p-value.


Rule: If the p-value is smaller than alpha, reject the null.


We have
p-value = 0.15866
alpha = 0.05
Therefore, we fail to reject the null.


Take note how the test statistic z = 1.00 is not to the right of the critical value z = 1.645, so we are not in the rejection region, and have further evidence to fail to reject the null.



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<font color=red>Summary:</font>
Null: p = 0.3
Alternative: p > 0.3
P-value: 0.15866  (approximate)
Critical value: z = 1.645 (approximate)
Decision: fail to reject the null (aka accept the null)



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Part (b)


Null: p = 0.75
Alternative: p < 0.75
This is a left-tailed test


The critical value is now z = -1.645 because we're doing a left-tailed test
P(Z < -1.645) = 0.05 approximately


Use a Z table to find that
P(Z < -2.50) = 0.00621
which is the approximate p-value.


The p-value is smaller than alpha = 0.05, so we will reject the null. Notice the test statistic is to the left of the critical value. The test statistic is in the rejection region.



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<font color=red>Summary:</font>
Null: p = 0.75
Alternative: p < 0.75
P-value: 0.00621  (approximate)
Critical value: z = -1.645 (approximate)
Decision: Reject the null




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Part (c)


Null: p = 0.375
Alternative: p =/= 0.375
The null will ALWAYS have the equal sign
In this case, the alternative hypothesis has the not equal sign (this is a two tailed test because of that).


Use a calculator or table to find the z critical values are roughly z = -1.960 and z = 1.960; a two-tailed test requires two critical values.
P(-1.960 < Z < 1.960) = 0.95 approximately
which can be rephrased as
P(Z < -1.960 or Z > 1.960) = 0.05 approximately


Rule: if the test statistic is between the critical values, then fail to reject the null. Otherwise, reject the null.


z = -1.94 is between -1.960 and 1.960
{{{
drawing(400,200,-3,3,-3,3,
line(-7,0,7,0),line(-2,0.04,-2,-0.04),line(-1,0.04,-1,-0.04),line(2,0.04,2,-0.04),

red(
locate(-2.1-0.5,-0.25,"-1.960"),locate(1.95,-0.25,"1.960"),
locate(-1,1.65,matrix(1,2,critical,values)),
line(-0.1,1.16,-1.66,0.3),line(-1.66,0.3,-1.4758,0.605),line(-1.66,0.3,-1.3038,0.293),
line(0.1,1.16,1.66,0.3),line(1.66,0.3,1.3038,0.293),line(1.66,0.3,1.4758,0.605)
),

blue(
circle(-1,0,0.06),
locate(-1.1,-0.25,"-1.94"),
locate(-1.1,-0.65,matrix(1,2,test,statistic))
),

locate(-3,-2,matrix(1,2,Diagram,not)),
locate(-3,-2.5,matrix(1,2,to,scale))
)
}}}
Therefore, we fail to reject the null.


Use a Z table to find that
P(Z < -1.94) = 0.02619
double this value because we're doing a two-tailed test
2*0.02619 = 0.05238


The p-value is roughly 0.05238
It is not smaller than alpha = 0.05, so this is further evidence we fail to reject the null.


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<font color=red>Summary:</font>
Null: p = 0.375
Alternative: p =/= 0.375
P-value: 0.05238  (approximate)
Critical values:  z = -1.960 and z = 1.960 (approximate)
Decision: Fail to reject the null
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