STATS-Business leaders around the world are becomi

Solution by Steps

step 1

We will conduct a chi-square test for independence to determine if there is a significant difference in the proportion of organizations that have embarked on digital transformation based on industry sector. The null hypothesis

H_0

states that there is no difference in proportions across sectors, while the alternative hypothesis

H_a

states that there is a difference

step 2

We will calculate the expected frequencies for each cell in the table using the formula:

E_{ij} = \frac{(row\ total_i)(column\ total_j)}{grand\ total}

where

E_{ij}

is the expected frequency for the

i

-th row and

j

-th column

step 3

The observed frequencies

O_{ij}

and expected frequencies

E_{ij}

will be used to calculate the chi-square statistic:

\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}

We will then compare this statistic to the critical value from the chi-square distribution with the appropriate degrees of freedom

step 4

The degrees of freedom for this test can be calculated as:

df = (r - 1)(c - 1)

where

r

is the number of rows and

c

is the number of columns in the contingency table. In this case,

df = (5 - 1)(2 - 1) = 4

step 5

Using a chi-square table or calculator, we will find the critical value for

df = 4

at the 0.05 significance level, which is approximately 9.488. If our calculated

\chi^2

exceeds this value, we will reject the null hypothesis

Answer

If the calculated chi-square statistic is greater than 9.488, we conclude that there is evidence of a difference in the proportion of organizations that have embarked on digital transformation based on industry sector.

Key Concept

Chi-square test for independence is used to determine if there is a significant association between two categorical variables.

Explanation

The chi-square test compares observed and expected frequencies to assess whether the distribution of responses differs significantly across categories.

Solution by Steps

step 1

To analyze the data from Table AMS 10.1, we will conduct an independent samples t-test to compare the update times of the two email interfaces. The null hypothesis

H_0

states that there is no difference in the mean update times between the two interfaces, while the alternative hypothesis

H_a

states that there is a difference

step 2

We will calculate the means and standard deviations for both email interfaces. Let

\bar{X_1}

and

\bar{X_2}

be the means for Email Interface 1 and Email Interface 2, respectively, and

s_1

and

s_2

be their standard deviations

step 3

The t-statistic can be calculated using the formula:

t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

where

n_1

and

n_2

are the sample sizes for each group (both are 15)

step 4

We will compare the calculated t-statistic to the critical t-value from the t-distribution table at a significance level of 0.05 with

n_1 + n_2 - 2

degrees of freedom. If the absolute value of the t-statistic is greater than the critical value, we reject the null hypothesis

step 5

For the second scenario, where each subscriber's update time is measured for both systems, we will conduct a paired samples t-test. The null hypothesis remains the same, but we will use the differences in update times for each subscriber to calculate the t-statistic

step 6

The t-statistic for the paired samples can be calculated using the formula:

t = \frac{\bar{D}}{s_D/\sqrt{n}}

where

\bar{D}

is the mean of the differences,

s_D

is the standard deviation of the differences, and

n

is the number of pairs (15)

Answer

The analysis will determine if there is a significant difference in update times between the two email interfaces based on the chosen statistical tests.

Key Concept

Independent samples t-test and paired samples t-test are used to compare means between two groups.

Explanation

The independent samples t-test is appropriate for comparing two different groups, while the paired samples t-test is used when the same subjects are measured under two conditions.

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