On average, are you receiving more messages than sending back? A random sample of 56 students shows the number of messages sent and received in one day. Numerical summaries for each variable and their difference are provided. Perform the test to check the claim that the number of messages received is higher than number of messages sent. Numerical summaries:
mean:
| Recieved | Sent | Difference | |
|---|---|---|---|
| Mean | 68.3 | 77.3 | 9 |
| St. Dev | 59.8 | 53.4 | 34.7 |
a. Explain, why the sample of differences between number of
messages received and sent is required?
b. What is the population parameter of interest?
c. Write the null and alternative hypotheses.
d. Perform the test at the significance level 0.05.
e. Estimate the p-value and make your decision.
f. State your conclusion in the terms of the problem.
a) We have to use Paired t test. since i) Both sample are dependent ii) sample sizes are equal
In Paired t test, we calculate difference (d) to second sample observation from first sample observation.
b) On average, the number of messages received is higher than number of messages sent .
c) Null Hypothesis: H0: the number of messages received is not higher than number of messages sent .
Alternative Hypothesis: H1: the number of messages received is higher than number of messages sent .
d) Test Statistic
t = d-bar / (SD / Sqrt(n))
= 9 / (34.7/sqrt(56)
= 1.9409
e) P-Value: 0.0287
f) since P-value < alpha 0.05, so we H0
Thus we conclude that there is sufficient evidence that the number of messages received is higher than number of messages sent .
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