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Probability Problems with Hypothesis Testing

1) Ms. Maria Wilson is considering running for mayor of the town of Bear Ghulch, Montana. Before completing the petitions, she decides to conduct a survey of voters in Bear Gulch. A sample of 400 voters reveas that 300 would support her in the November election.

a) estimate the value of the population proportion.
b) compute the standard error of the prportion.
c) develop a 99 percent confidence interval for the population proportion
d) interpret your findings

2)The postanesthesia care area (recovery room) at St. Luke's Hospital in Maumee, Ohio, was recently enlarged. The hope was that with the enlargement the mean number of patients per day would be more than 25. A random sample of 15 days reveals the follwing numbers of patients. 25 27 25 26 25 28 28 27 24 26 25 29 25 27 24

At the .01 significance level, can we conclude that the mean number of patients per day is moire than 25? Estimate the p-value and interpret it.

3)A nationwide sample of influetial Republicans and Democrats was asked as a part of a comprehensive survey whether they favored lowering environmental standards so that high-sulfur coal could be burned in coal-fired power plants. The result were:

Republicans Democrats

Number sampled 1,000 800
Number in favor 200 168

At the .02 level of significance, can we conclude tht there is a larger proportion of Democrats in favor of lowering the standards?

4) ) The null and alternative hypotheses are :

H0: μσ = 0
H1: μσ ≠ 0

The following paired observations show the number of traffic citations given for speeding by Officer Dhondt and Officer Meredith of the South Carolina Highway Patrol for the last five months.

May June July August September

Officer Dhondt 30 22 25 19 26
Officer Meredith 26 19 20 15 19

At the .05 significance level, is there a difference in the mean number of citations given by the two officers?

Note: Use the five-step hypothesis testing procedure to solve the following exercises

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Here are your answers.

Question 1
Part A
We're told that 300 out of the 400 voters in the sample support Maria Wilson. The sample proportion is thus 300/400 = 0.75. Since the best estimate for the population proportion is the standard proportion, then we estimate the population proportion at 0.75

Part B
The standard error of the proportion can be found using the following formula:
Std Error = sqrt(p*(1-p)/n)
[ sqrt() means "square root of"]

where p is the proprotion, and n is the sample size. Therefore, the standard error is:
sqrt(0.75*0.25/400) = 0.00046875

Part C and D
By virtue of the Central Limit Theorem, we know that this sample proportion should follow approximately a normal distribution, with mean equal to the population proportion, and the standard error we calculated above. Therefore, what we must do here is calculate an interval, centered in 0.75 (the sample proportion) such that 99% of the observations from a normal distribution are contained within it. The formula for this interval is:

[ p - z(0.005)*StdError , p + z(0.005)*StdError]
where p is the proportion (0.75), StdError is the standard error we obtained earlier and z(0.005) is the value of standard normal distribution such that values greater than it have a probability of 0.005.

Looking up in a table, we find that this z-value is 2.57. Therefore, the interval is:
[ 0.75 - 2.57*0.00046875, 0.75 + 2.57*0.00046875 ]
= [0.7487, 0.7512]

So, with 99% confidence, the population proportion is within [0.7487, 0.7517]. This implies that Maria Wilson will almost surely win the election, as there is a 99% probability that she will obtain at least 74.87% of the votes.

Question 2
First of all, let's write the null and alternative hypothesis:

H0: μ; >= 25
Ha: ...

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