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Duke Energy improperly disposed of chemicals ( coal ash ) which have leaked into the ground water along the Dan River. If less than 5% of the water supply contains coal ash, the water is considered safe. Officials believe the water in Greensboro may be unsafe. They select samples of water from 200 locations in the Greensboro area and record whether or not coal ash is present. They find that 8% of the sampled water supply contains coal ash.
State appropriate hypotheses for performing a significance test using words and symbols.
H0: p = 5% Ha: p > 5%
After conducting a significance test, a P value of 0.026 is found. interpret this value
there is an 0.026 % chance of 200 random samples of water having 8% coal ash assuming h0 is true.
Based on the P - value and a significance level of 0.05 should greensboro keep the current water or switch to bottled water? Explain
Greensboro should switch to bottled water, as 0.026 is less than 0.05.
Let’s suppose this decision is wrong. What would be a consequence of this error?
Overspending on bottled water and contributing to plastic pollution.
Given the water is safe, how often will this error occur? 2.6% of the time, assuming the math was done correctly.
Suppose we use a significance level of 0.01. Based on the p-value and new alpha, should the Greensboro area keep the current water or switch to bottled water? Explain.
The Greensboro area should keep the current water, as .01 is less than .026
Lets suppose this decision is wrong. What would be a consequence of this error ?
People would drink water which might make them sick. They might die.
Are the consequences in question #4 or #7 more serious? Explain.
The consequences from question #7 ( illness or death for population ) are more serious than the consequences from question #4 ( overspending and polluting )
Important Ideas : Type 1 & Type 2 Errors and Power
water is safe
water is unsafe
h0: p < 5% is not necessary we can only test a null hypothesis of equal to even if we would prefer it to be <
p = .08
^
Hey: we did everything right, this is our final recommendation, however, it is possible this recommendation is wrong, and if this is wrong, here is what it would mean for you. You need to be able to do all the things and step back and say hey but if were wrong this is what would happen
This is called a type 1 error H0 is true BUT we reject
5% - significance level. Significance level is how often we are gonna be wrong.
Side Detour: significance level is how unlikely it needs to be in order for it to be a coincidence.
This is called Type 2:
Type 2 generally happens when h0 is false BUT we fail to reject.
Depends on context
Truth
Decision
h0 true
h0 false
reject h0
fail to reject h0
Yay! we were right!!
Yay! we were right!!
Type 1
Type 2
Type 2 is fail 2 reject
Type 1: The H0 is true, but we do have convincing evidence for Ha.
Type 2: The H0 is false, but we dont have convincing evidence for Ha.
P(Type 1) = significance level
P(Type 2) = 1 - Power
We will be given power or the info required to find power
Check Your Understanding!
Mr Wilcox purchased a trick coin that is supposed to land heads up 75% of the time. One of his students volunteer to test this claim. The student flips the coin 50 times and finds the coin lands heads up 35 times. The student then performs a test of the following hypotheses at the significance level = 0.10 significance level.
H0 : p = 0.75 Ha : p < 0.75
where p = the true proportion of the tosses of this coin that would land heads up
H0: the coin lands heads up 75% of the time Ha: the coin lands heads up less than 75% of the time
- Describe a Type 1 error and a Type 2 error in this setting:
A type 1 error would be us not having convincing evidence for h0, but h0 being true. So we think it is less than 75% of coins when infact it is 75% of the time heads up.
A type 2 error would be us thinking that it was landing heads up 75% of the time, when in fact it is landing heads up less than 75% of the time.
- Which type of error may result in Mr. Wilcox returning the coin and writing a negative review of the product
Type 1
- If the student were to use a = 0.05 instead of a = 0.10 would this make it more or less likely to reject the null hypothesis when the null hypothesis is true ? explain
if the student used a higher significance level type 1 would be higher, as the probability of a type 1 error is equal to the significance level
- The probability of committing a type 2 error is determined to be 0.15. Find the power of the test
1-.15 = .85
power is .85
P(Type 2 ) = 1 - power