Question

Environmental engineers are using data collected by weather data centers to learn how climate affects the sea ice. Of 511 ice melt ponds studied in a certain​region, 80 were classified as having​ "first-year ice". The researchers estimated that about 16​% of melt ponds in the region have​ first-year ice.​ Estimate, with​ 90% confidence, the percentage of all​ ice-melt ponds in the region that have​ first-year ice. Give a practical interpretation of the results.

Construct a​ 90% confidence interval around the sample proportion of ice melt ponds with​ first-year ice.

left parenthesis nothing comma nothing right parenthesis,

​(Round to four decimal places as​ needed.)

Interpret the confidence interval practically. Choose the correct answer below.

A.Since 16​% is in the​ interval, one can be​ 90% confident the true proportion of ice melt ponds in the region with​ first-year ice is 16​%.

B.One can be​ 90% confident the true proportion of ice melt ponds in the region with​ first-year ice is within the above​ interval, and there is a​ 90% chance it is 16​%.

C.One can be​ 90% confident the true proportion of ice melt ponds in the region with​ first-year ice lies at the mean of the above​ interval, rather than at 16​%.

D.One can be​ 90% confident the true proportion of ice melt ponds in the region with​ first-year ice is within the above​ interval, though it is probably not 16​%.

E.Since 16​% is not in the​ interval, one can be​ 90% confident the true proportion of ice melt ponds in the region with​ first-year ice is not 16​%.

Answer 1

Sol:

Ho:p=16%=0.16

Ha:p not =0.16

Z crit for 90%=1.645

90% confidence interval for the percentage of all​ ice-melt ponds in the region that have​ first-year ice.

p^-z*sqrt(p^*(1-p^)/n,p^+z*sqrt(p^*(1-p^)/n

p^=sample proportion=x/n=80/511=0.1565558

z crit=1.645

n=511

so

0.1565558-1.645*sqrt(0.1565558*(1-0.1565558)/511),0.1565558+1.645*sqrt(0.1565558*(1-0.1565558)/511)

0.1301,0.18299

0.1301*100,0.183*100

13.01%,18.3%

lower 90%=13.01%

upper 90%=18.3%

13.01<16<18.3

Fail to reject H0

Accept Ho.

A.Since 16​% is in the​ interval, one can be​ 90% confident the true proportion of ice melt ponds in the region with​ first-year ice is 16​%.