Complete Question
Environmental engineers are using data collected by weather data centers to learn how climate affects the sea ice. Of 526 ice melt ponds studied in a certain region, 84 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.
Answer:
The 90% confidence interval is [tex] 0.1337 < p < 0.1857 [/tex]
Step-by-step explanation:
From the question we are told that
The sample size is n = 526
The number that were classified to having a 'first-year ice ' is k = 84
The population proportion is p = 0.16
Generally the sample proportion is mathematically represented as
[tex]\^ p = \frac{k}{ n }[/tex]
=> [tex]\^ p = \frac{ 84}{ 526 }[/tex]
=> [tex]\^ p = 0.1597[/tex]
From the question we are told the confidence level is 90% , hence the level of significance is
[tex]\alpha = (100 - 90 ) \%[/tex]
=> [tex]\alpha = 0.10[/tex]
Generally from the normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.645[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \sqrt{\frac{ p (1- p)}{n} } [/tex]
=> [tex]E = 1.645 * \sqrt{\frac{0.16 (1- 0.16)}{ 526} } [/tex]
=> [tex]E = 0.026 [/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\^ p -E < p < \^ p +E[/tex]
=> [tex]0.1597 -0.026 < p < 0.1597 -0.026 [/tex]
=> [tex] 0.1337 < p < 0.1857 [/tex]
