Answer:
90% confidence interval for the true mean % cacao is [53.87% , 56.13%].
Step-by-step explanation:
We are given that in order to determine the mean % cacao in its dark chocolate products, quality inspectors sample 36 pieces.
They find a sample mean of 55% with a standard deviation of 4%.
Firstly, the pivotal quantity for 90% confidence interval for the true mean is given by;
P.Q. = ~
where, = sample mean % cacao = 55%
s = sample standard deviation = 4%
n = sample of pieces = 36
= true mean % cacao
<em>Here for constructing 90% confidence interval we have used One-sample t test statistics as we know don't about population standard deviation.</em>
So, 90% confidence interval for the true mean, is ;
P(-1.6895 < < 1.6895) = 0.90 {As the critical value of t at 35 degree of
freedom are -1.6895 & 1.6895 with P = 5%}
P(-1.6895 < < 1.6895) = 0.90
P( < < ) = 0.90
P( < < ) = 0.90
<em><u>90% confidence interval for</u></em> = [ , ]
= [ , ]
= [0.5387 , 0.5613]
= [53.87% , 56.13%]
Therefore, 90% confidence interval for the true mean % cacao is [53.87% , 56.13%].