# Two probability and hypothesis testing questions

Question 5: The weight of bags "checked in" is normal, with a mean of 30 pounds and a standard deviation of 9 pounds.

a) What is the probability that a randomly selected bag will weigh more than 48 pounds?

b) What is the probability that the total weight of 4 bags will be more than 192 pounds?

c) Find the probability that a sample of 81 bags will average more than 32 pounds.

d) If a sample of 36 bags showed the mean to be 33 pounds, can you conclude that the average weight has changed? (use alpha = 0.05)

Question 6: Test the hypothesis that more than 60 percent of the U.S. senators (of a total 100 senators) favor the president's economic plan, if a sample of 40 showed that 22 favor the plan (use alpha = .05).

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Question 5

The weight of bags "checked in" is normal, with a mean of 30 pounds and a standard deviation of 9 pounds.

a) What is the probability that a randomly selected bag will weigh more than 48 pounds?

For this problem, we're going to use z-scores. To calculate a z-score, use the following formula:

.

where x is the score to be converted into a z-score, σ is the standard deviation of the population, and μ is the mean of the population

For this problem, μ = 30, σ = 9, and x = 48. Calculate the z-score:

z = (48 - 30)/9 = 18/9 = 2

Now, look at a z-distribution table to see the probability associated with this z-score. This is equivalent to finding the area under the z-distribution curve to the right of z = 2. You should find that the p-value/area under the curve is equal to p = 0.0228.

This means that the probability that a randomly selected bag will weigh more than 48 pounds is 0.0228 or 2.28%.

b) What is the probability that the total weight of 4 bags will be more than 192 pounds?

Now that we have a sample size, we can use a different formula for ...

Probability and Hypothesis testing

1. Suppose that the mean of the annual return for common stocks from 2000 to 2012 was 14.37%, and the standard deviation of the annual return was 35.14%. Suppose also that during the same 12-year time span, the mean of the annual return for long-term government bonds was 0.6%, and the standard deviation was 2.1%. The distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric in this scenario. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.

a Find the probability that the return for common stocks will be greater than 16.32%.

b. Find the probability that the return for common stocks will be greater than 5.89%.

c. Find the probability that the return for common stocks will be less than 14.37%.

2. The management of a computer software company is considering relocating the corporate office (A) to a new location out of state, office (B). The management is concerned that the commute times of the employees to the new office (B) might be too long. The company decides to survey a sample of employees at other companies in the same office (A) to see how long these employees are commuting to the office. A sample of 23 employees indicated that the employees are commuting X (bar) = 33 minutes and s = 1 minute, 45 seconds.

1. a. Using the 0.01 level of significance, is there evidence that the population mean is above 32 minutes?

2. b. What is your answer in (a) if X (bar) = 37 minutes and s = 27 minutes?

3. c. Look at your answers for a and b above and discuss what you can learn from the results about the effect of a large standard deviation.