A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of awriting

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A certain pen has been designed so that true average writing lifetime under controlled conditions (involving the use of awriting

machine) is at least 10 hours. A random sample of 18 pens is selected, the writing lifetime of each is determined,and a normal probability plot of the resulting data supports the use of a one-sample t test.a. What hypotheses should be tested if the investigators believe a priori that the design specification has been satisfied?b. What conclusion is appropriate if the hypotheses of part (a) are tested,t = —23, and a = .05?c. What conclusion is appropriate if the hypotheses of part (a) are tested, t = —1.8, and u = .01?d. What should be concluded if the hypotheses of part (a) are tested and t = —3.6 ?

Mathematics

1 answer:

tester [12.3K] · Answer 1 · 2026-02-26T23:41:47+03:00

Answer:

a) Null hypothesis: $\mu \geq 10$

Alternative hypothesis: $\mu < 10$

b) $p_v =P(t_{17}$

Given that the p-value is lower than the significance threshold in this situation, we have sufficient grounds to reject the null hypothesis.

c) $p_v =P(t_{17}$

In this case, since the p-value exceeds the significance threshold, we have adequate evidence to FAIL to reject the null hypothesis.

d) $p_v =P(t_{17}$

Here again, with the p-value being less than the significance level, we can reject the null hypothesis.

Step-by-step explanation:

1) Provided data and references

$\bar X$ represents the average of the samples

$s$ denotes the standard deviation of the samples

$n=18$ indicates the number of samples

$\mu_o =10$ is the value we are examining

$\alpha$ defines the significance level for the test.

t represents the specific statistic of interest

$p_v$ indicates the p-value relevant to the test (the variable of concern)

Define the null and alternative hypotheses.

To assess if the true mean is at least 10 hours, we must set up a hypothesis:

Part a

Null hypothesis: $\mu \geq 10$

Alternative hypothesis: $\mu < 10$

If we consider the sample size being less than 30 and the population deviation unknown, it’s more appropriate to use a t-test to compare the actual mean with the reference value, calculated as:

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$ (1)