The data below are duration times, in minutes, for the Old Faithful geyser from

Question

The data below are duration times, in minutes, for the Old Faithful geyser from two different years.

Old faithful Eruption Durations (minutes)

January 8,2000

3.37 3.87 4.00 4.03 3.50 4.08 2.25 4.70 1.73 4.93 1.73 4.62 3.43 3.25

1.68 3.92 3.68 3.10 4.03 1.77 4.08 1.75 3.20 1.85 4.62 1.97 4.50 3.92

4.35 2.33 3.83 1.88 4.60 1.80 4.73 1.77 3.57 1.85 3.52 4.00 3.70 3.72

3.25 3.58 3.80 3.77 3.75 2.50 4.50 4.10 3.70 3.80 3.43 4.00 2.27 4.40

January 8,2010

4.05 4.25 3.33 2.99 4.33 2.93 4.58 1.90 3.58 3.73 3.73 1.82 4.63 3.50

4.00 3.67 1.67 4.60 1.67 4.00 1.80 4.42 1.90 4.63 2.93 3.50 1.97 4.28

4.83 4.13 1.93 4.65 4.20 3.93 4.33 4.83 4.53 2.03 4.18 4.43 4.07 4.13

3.95 4.10 2.27 4.58 4.90 4.50 1.95 4.83 4.12

Use appropriate hypothesis tests for a, b.

a) It is thought that the duration length of the geyser is chaning with time. Can you conclude that there is a difference in the mean duration lengths for the two dates?

So, I know I need t find the z score using formula Z= (X-Y - Delta-mu) / (sqrt(sigX/Nx + sigY/Ny))

b) Some believe the geyser is getting more reliable. Can you conclude that the variance in 2010 is less than the variance in 2000?

So this one needs to use F value, mainly need help with Null and alternate hypothesis as I always get them wrong

Explanation / Answer

Solution:

Here, we have to use the two sample t test for the population mean because we don’t know the population mean for the duration length of the geyser. We have to check the following null hypothesis by using the two sample t test.

Null hypothesis: H0: There is no any significant difference in the mean duration lengths for the two dates.

Alternative hypothesis: Ha: There is a significant difference in the mean duration lengths for two dates.

We assume the level of significance for this test as 0.05 or 5%. The test is given as below:

Calculations Area

Pop. 1 Sample Variance

0.9604

Pop. 2 Sample Variance

1.0692

Pop. 1 Sample Var./Sample Size

0.0172

Pop. 2 Sample Var./Sample Size

0.0210

For one-tailed tests:

TDIST value

0.1025

1-TDIST value

0.8975

Separate-Variances t Test for the Difference Between Two Means

(assumes unequal population variances)

Data

Hypothesized Difference

0

Level of Significance

0.05

Population 1 Sample

Sample Size

56

Sample Mean

3.393928571

Sample Standard Deviation

0.9800

Population 2 Sample

Sample Size

51

Sample Mean

3.642941176

Sample Standard Deviation

1.0340

Intermediate Calculations

Numerator of Degrees of Freedom

0.0015

Denominator of Degrees of Freedom

0.0000

Total Degrees of Freedom

102.7528

Degrees of Freedom

102

Standard Error

0.1952

Difference in Sample Means

-0.2490

Separate-Variance t Test Statistic

-1.2755

Two-Tail Test

Lower Critical Value

-1.9835

Upper Critical Value

1.9835

p-Value

0.2050

Do not reject the null hypothesis

Here, we have to use the F test for the differences in the two variances for the data for 2010 and 2000. The F test is given as below:

F Test for Differences in Two Variances

Data

Level of Significance

0.05

Larger-Variance Sample

Sample Size

51

Sample Variance

1.069245176

Smaller-Variance Sample

Sample Size

56

Sample Variance

0.960417013

Intermediate Calculations

F Test Statistic

1.1133

Population 1 Sample Degrees of Freedom

50

Population 2 Sample Degrees of Freedom

55

Upper-Tail Test

Upper Critical Value

1.5774

p-Value

0.3478

Do not reject the null hypothesis

Calculations Area

Pop. 1 Sample Variance

0.9604

Pop. 2 Sample Variance

1.0692

Pop. 1 Sample Var./Sample Size

0.0172

Pop. 2 Sample Var./Sample Size

0.0210

For one-tailed tests:

TDIST value

0.1025

1-TDIST value

0.8975

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