Question

1. What are type I and type II errors? 2. How do we determine the critical numbers and/or critical regions for a hypothesis test? 3. Explain when to use a 2-test, a t-test, a x-test, and an F-test. 4. How does decreasing the significance level affect the probability of a type I error? 5. If the correlation between two variables is significant, does one cause the other? 6. What does measure? 7. A real estate agent believes that the average closing cost of purchasing a new home is $6500 over the purchase price. She selects 40 new home sales at random and finds that the average closing costs are $6600. The standard deviation is $120. Test her claim at the 5% significance level a) State the null and alternative hypotheses

Answer 1

1)

Type I error -

“A Type I error is the probability of rejecting the null hypothesis when the null hypothesis is actually TRUE”. It is the probability of correctly rejecting the null hypothesis. Type I error can be defined by significance level .

Type II error -

“A Type II error is the probability of failing to reject the null hypothesis when the null hypothesis is actually FALSE. Type II error is denoted by .

2)

The critical numbers and/or critical regions for a hypothesis test is defined by significance level .

3)

Z-test: To compare means when population standard deviation is known

t-test: To compare means when population standard deviation is unknown

Chi-square: To compare the expected frequencies (or proportion) with the observed frequencies (or proportion) data values.

F-test: To compare the variance of two sample.

4)

Since type I error is also be defined by significance level , If decrease one other will also decrease.

5)

Correlation doesn't mean causation. It doesn't tell the cause of variation in one variable is due to other variable.

6)

The R square value measure the percentage variation in dependent variable, Y which is explained by the independent variable, X.

7)

a)

T-test for One Population Mean is used to compare the sample mean with hypothesize population mean.

The Null and Alternative Hypotheses

Η :μ = 6500

$H_1: \mu\neq 6500$

b)

The t-critical value is obtained from t distribution table for degree of freedom = n - 1 = 40 - 1 = 39 and significance level = 0.05

tc, upper = 2.023

tc,lower = -2.023

c)

Left Tail Two-Tail Right Tail 0.40 0.35 0.30 0.25 0.025 0.950 0.025 0.15 0.10 0.05 0.00 -4 -1 0 b -2.023 2.023

d)

The t statistic is obtained using the formula,

From the data values,

$t=\frac{6600-6500}{20/ \sqrt{40}}$

$t=5.27$

e)

The t statistic is greater than t-critical value, the null hypothesis is rejected.

8)

a)

T-test for One Population Mean is used to compare the sample mean with hypothesize population mean.

The Null and Alternative Hypotheses

$H_0: \mu=12.4$

H, :μ + 12.4

b)

The t-critical value is obtained from t distribution table for degree of freedom = n - 1 = 15 - 1 = 14 and significance level = 0.01

$t_{c,upper}=2.977$

to lower = -2.977

c)

Left Tail Two-Tail Right Tail 0.40 0.35 0.30 0.25 0.20 0.0050 0.990 0.0050 0.15 0.10 0.05 0.00 -1 0 1 -2.977 2.977

d)

The t statistic is obtained using the formula,

From the data values,

$t=\frac{11.2-12.4}{2/ \sqrt{15}}$

t = -2.324

e)

The t statistic is less than t-critical value, the null hypothesis is not rejected.