Given:
�
1
=
367
,
�
1
=
535
,
�
2
=
436
,
�
2
=
562
,
confidence level
=
90
%
x
1
=367,n
1
=535,x
2
=436,n
2
=562,confidence level=90%
First, calculate the sample proportions:
�
^
1
=
�
1
�
1
=
367
535
p
^
1
=
n
1
x
1
=
535
367
�
^
2
=
�
2
�
2
=
436
562
p
^
2
=
n
2
x
2
=
562
436
Next, calculate the standard error:
SE
=
�
^
1
(
1
−
�
^
1
)
�
1
+
�
^
2
(
1
−
�
^
2
)
�
2
SE=
n
1
p
^
1
(1−
p
^
1
)
+
n
2
p
^
2
(1−
p
^
2
)
Then, find the critical value for a 90% confidence level. Since the confidence level is given as 90%, the corresponding two-tailed critical value is
�
=
1.645
z=1.645 (obtained from a standard normal distribution table).
Finally, plug the values into the formula to calculate the confidence interval:
Confidence Interval
=
(
(
�
^
1
−
�
^
2
)
±
�
⋅
SE
)
Confidence Interval=((
p
^
1
−
p
^
2
)±z⋅SE)
Let's calculate it step by step:
�
^
1
=
367
535
≈
0.686
p
^
1
=
535
367
≈0.686
�
^
2
=
436
562
≈
0.775
p
^
2
=
562
436
≈0.775
SE
=
0.686
(
1
−
0.686
)
535
+
0.775
(
1
−
0.775
)
562
≈
0.034
SE=
535
0.686(1−0.686)
+
562
0.775(1−0.775)
≈0.034
Confidence Interval
=
(
(
0.686
−
0.775
)
±
1.645
⋅
0.034
)
Confidence Interval=((0.686−0.775)±1.645⋅0.034)
Now, calculate the upper and lower bounds of the confidence interval:
Lower bound
=
(
0.686
−
0.775
)
−
1.645
⋅
0.034
Lower bound=(0.686−0.775)−1.645⋅0.034
Upper bound
=
(
0.686
−
0.775
)
+
1.645
⋅
0.034
Upper bound=(0.686−0.775)+1.645⋅0.034
Rounding the values to three decimal places, the confidence interval is approximately:
Confidence Interval
=
(
−
0.102
,
−
0.065
)
Confidence Interval=(−0.102,−0.065)
Therefore, the researchers are 90% confident that the difference between the two population proportions,
�
1
−
�
2
p
1
−p
2
, is between -0.102 and -0.065 (in ascending order).