Statistics

Statistics

1. Asuming that the following sample of 10 numbers arise from a Normal distribution with mean µ and standard deviation s = 6.0, obtain, from first principles, 95% and 99% confidence intervals for µ.

39.4 22.1 37.6 26.4 26.5 29.6 25.4 30.9 28.3 22.4

Evaluate a test statistic for the null hypothesis H0: µ = 30 against the two-sided alternative hypothesis H1 : µ = 30, and by plotting this on its reference distribution, identify and calculate the corresponding p-value associated with H0. What are your conclusions about H0?

One-Sample Statistics

N

Mean

Std. Deviation

Std. Error Mean

VAR00001

10

28.8600

5.80731

1.83643

One-Sample Test

Test Value = 30

t

df

p-value

Mean Difference

95% Confidence Interval of the Difference

Lower

Upper

VAR00001

-.621

9

.550

-1.14000

-5.2943

3.0143

Comments:

Since calculated value is greater than the tabulated value therefore we reject Ho.

2. Repeat all calculations for the confidence intervals for µ and the p-value for H0 for the data in Q1 in the absence of any assumption on value of the standard deviation s. Do your conclusions about H0 change?

One-Sample Statistics

N

Mean

Std. Deviation

Std. Error Mean

VAR00001

9

29.5778

5.66961

1.88987

One-Sample Test

Test Value = 30

t

df

Sig. (2-tailed)

Mean Difference

95% Confidence Interval of the Difference

Lower

Upper

VAR00001

-.223

8

.829

-.42222

-4.7803

3.9358

Comments:

From above calculation we noticed that value of Ho changes as per change in data of assumption.

3. Tables show that the 95% critical values for some t distributions are (to 2 dp) as follows:

Df69 12 20 30

critical values ±2.45 ±2.26 ±2.18 ±2.09 ±2.04

Use statistical tables or the TINV function in EXCEL to find the critical values for the t distribution with the following degrees of freedom

5 8 15 18 24 60 120

Combine the two tables, and hence argue that these critical values approach ±1.96 as the number of degrees of freedom tends to infinity.

Briefly comment on why we would should expect ±1.96 to be the limit here.

Investigate the behavior of the 99% critical values for these degrees of freedom, and comment on the limits in this case.

0.0130643790.0128858350.012797060.0126910810.012638349

Comments:

From above table we noticed that TINV decreases as degree of freedom increases.

4. Suppose that n = 9, ¯X = 85 and the sample standard deviation is

s = 2. Test H0 : µ = 83 against H1 : µ = 83, stating any assumptions made about the underlying population, and the testing procedure.

Comments:

The observed t-value is 3.

Using the .05 level of significance, reject the hypothesized value.

5. A random sample of 10 flights from Amsterdam to Paris yields the following flight-times (in minutes):

40 45 43 36 38 41 38 43 37 39

Determine a 95% confidence interval for µ, and use this to discuss ...