Your professor claims that half the beads in an urn are red

Your professor claims that half the beads in an urn are red. A random sample of 100 beads is selected, 39 of which are red. Calculate a 95% confidence interval for the proportion of red beads in the urn. (round the margin of error to two decimal places)

                                                                  ? X ?

2. If a coffee roaster determines that more than 7% of the beans in its current shipment are unusable, then the shipment will be sent away and another brought in. A supervisor selects a random sample of 350 beans from the current shipment and finds that 30 are not yet ripe enough and must be discarded (the rest of the beans in the sample are fine). Carry out a significance test at the =0.1 significance level with correction and determine whether or not the roaster will send the shipment back or not.

     a. The roaster will send the shipment back
     b. The roaster will keep the shipment

3. Suppose you want to estimate how many red beads are inside an urn, and you suspect that half of the beads in the urn are actually red. If your suspicion is true, how many beads would you have to sample to estimate the proportion within a margin of error of 0.05 with 95% confidence?

4. Two surveys were conducted to determine the effect that gender may play in favoring a certain political candidate. In the first survey, which was exclusively for women, 56% of the 251 respondants said they would vote for the candidate. Meanwhile, 43% of the 330 male respondants in the second survey said that they would vote for the candidate.

a) Find a 95% confidence interval for the difference of proportions.
                                          ? X ?

b) Can we conclude, at 95% significance, that the proportions differ?
                               Yes or No

Solution

1.

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.39          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.048774994          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
Margin of error = z(alpha/2)*sp =    0.095597231 = 0.10          
lower bound = p^ - z(alpha/2) * sp =   0.294402769          
upper bound = p^ + z(alpha/2) * sp =    0.485597231          
              
Thus, the confidence interval is              
              
(   0.29   ,   0.49   ) [ANSWER]

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