From her firm’s computer telephone log, an executive found that the mean length of 64 telephone calls during July was 4.48 minutes with a standard deviation of 5.87 minutes. She vowed to make an effort to reduce the length of calls. The August phone log showed 48 telephone calls whose mean was 2.396 minutes with a standard deviation of 2.018 minutes. (a) State the hypotheses for a right-tailed test. (b) Obtain a test statistic and p-value assuming unequal variances. Interpret these results using á = .01. (c) Why might the sample data not follow a normal, bell-shaped curve? If not, how might this affect your conclusions?
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(a) The hypotheses for a right-tailed test are:
- Null hypothesis (H0): The mean length of telephone calls in August is greater than or equal to the mean length of telephone calls in July.
- Alternative hypothesis (Ha): The mean length of telephone calls in August is less than the mean length of telephone calls in July.
(b) To obtain the test statistic and p-value assuming unequal variances, we can use the t-test formula:
t = (x1 - x2) / √[(s1^2/n1) + (s2^2/n2)]
Where:
x1 = mean length of telephone calls in July = 4.48 minutes
x2 = mean length of telephone calls in August = 2.396 minutes
s1 = standard deviation of telephone calls in July = 5.87 minutes
s2 = standard deviation of telephone calls in August = 2.018 minutes
n1 = number of telephone calls in July = 64
n2 = number of telephone calls in August = 48
Plugging in the values:
t = (4.48 - 2.396) / √[(5.87^2/64) + (2.018^2/48)]
(c) The sample data might not follow a normal, bell-shaped curve if there are outliers, the data is skewed, or if the population of telephone calls has a non-normal distribution. If the sample data does not follow a normal distribution, it could affect the conclusions because the t-test assumes a normal distribution. In this case, the validity of the test results may be compromised and alternative statistical tests or approaches may need to be used.
(a) The hypotheses for a right-tailed test can be stated as follows:
Null hypothesis (H0): The mean length of telephone calls in August is not significantly shorter than the mean length of telephone calls in July.
Alternative hypothesis (Ha): The mean length of telephone calls in August is significantly shorter than the mean length of telephone calls in July.
(b) To obtain a test statistic and p-value assuming unequal variances, we can use the independent samples t-test.
The test statistic can be calculated using the formula:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.
In this case, x1 = 4.48 minutes, x2 = 2.396 minutes, s1 = 5.87 minutes, s2 = 2.018 minutes, n1 = 64, and n2 = 48.
Calculating the test statistic:
t = (4.48 - 2.396) / sqrt((5.87^2 / 64) + (2.018^2 / 48))
= 2.084
To obtain the p-value, we need to consult the t-distribution table or use statistical software. Assuming a significance level (alpha) of 0.01, we compare the test statistic to the critical value from the t-distribution. If the test statistic is greater than the critical value, we reject the null hypothesis.
(c) The sample data might not follow a normal, bell-shaped curve due to various reasons such as outliers, small sample sizes, or non-normal underlying population distributions. If the data does not follow a normal distribution, it may affect the validity of the t-test results.
In this case, if the sample data does not follow a normal distribution, the conclusions drawn from the t-test might not be reliable. It is important to consider potential biases or limitations in the analysis and interpret the results with caution. Further assessments such as conducting robustness checks or alternative statistical tests may be necessary.