# Regression ,Hypotheses testing and confidence intervals

1. Scores on a statistics test are normally distributed with a mean of 80 and a SD of 5. The average for a class of 30 is 90. The teacher of that class says that her class has done significantly better than average.

a. What null hypothesis would you use to test the teacher's statement?

b. What is the mean and the standard deviation of the distribution you would use to test that hypothesis?

c. What is the p-value? What conclusion can you draw from it?

2. You have 10 gifts; they include 2 dolls, 2 trucks, 3 books, and 3 balls. Children are asked to choose a gift, without knowing what they are choosing.
a. What is the probability that the first child picks a truck and the second picks a doll?

b. What is the probability that the first child picks a book or a doll?

3. Calculate the area for the following values of X, when the mean is 10 and the standard deviation is 2.
a. P(X<8)

b. P(X>14)

c. P(8<X<14)

4. As your sample size increases
a. the p-value will increase
b. the mean will increase
c. the standard deviation will decrease
d. the chance of accepting the null hypothesis will increase

5. Explain the two types of mistakes you can make in hypothesis testing. Give an example when you would be more concerned about one than the other.
.

6. X has a uniform distribution between 0 and 4.
a. Draw the distribution.

b. Find P(X< 2)

c. What is P( 3<X<4)?

7. The following table includes men's and women's records in the 800-meter run in selected years.

Year Men's Women's record
1905 113.4 ---
1915 111.9 ----
1925 111.9 144.0
1935 109.7 135.6
1945 106.6 132.0
1955 105.7 125.0
1965 104.3 118.0
1975 104.1 117.5
1985 101.73 113.28
1995 101.73 113.28

a. Do a scatter plot of time, depending on years (this will be easier if you use only the last two digits of the year, e.g. 05, 15, 25,...) for both men and women.

b. Describe the plots. Are there any outliers? Is there a difference between the men and the women?

c. Find the equations for both sets of data.

d. What do expect the times to be in the year 2010?

8. A machine bottling soda overfills bottles 0.1% of the time. An inspector watches a sample of bottles and finds that the machine overfilled 5 out of 1000 bottles.
a. Is there a reason to believe the machine is not working correctly?

b. What is a 95% confidence interval for the sample value of p (5 in 1000, or 0.5%)?

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#### Solution Preview

...ck.
P(A) = 2/10
Let B be the event that the second child picks a doll.
We need P(B|A) = 2/9
Thus the probability that t the first child picks a truck and the second picks a doll
Is P(AB) =P(A).P(B|A) =( 2/10)*(2/9) =4/90
b. What is the probability that the first child picks a book or a doll?
A : Event of selecting a book
B: Event of selecting a doll
We need P(AB) = P(A)+P(B) =2/10+3/10=5/10

3. Calculate the area for the following values of X, when the mean is 10 and the standard deviation is 2.
Given that X is normal with mean 10 and standard deviation 2. Standardizing the variable using Z score and from normal tables we have
a. P(X<8)
P(X<8) = =P(Z<-1) =0.1586
b. P(X>14)
P(X > 14) = =P(Z>2) =0.0228
c. P(8<X< 14)
= P(-1<Z<2) =0.8196
4. As your sample size increases
a. the p-value will increase
b. the mean will increase
c. the standard deviation will decrease
d. The chance of accepting the null hypothesis will increase
The standard deviation of the sample mean is given by . Thus when n increase , the sample standard deviation decrease .

5. Explain the two types of mistakes you can make in hypothesis testing. Give an example when you would be more concerned about one than the other.
Type I error
A type I error occurs when one rejects the null hypothesis when it is true. It occurs when we are observing a difference when in truth there is none.
Type II error
A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true. This is ...