Question

A manufacturer claims that the mean lifetime of its lithium batteries is 1100 hours buta group of homeowners believe their battery-life is different than this manufactureds claim. A homeowner selects 35 of these batteries and finds the mean lifetime to be 1080 hours with a standard deviation of 80 hours. Test the manufacturer's claim. Use u = 0.10 and the p-value approach. Round the test statistic to the nearest thousandth.

Every solution should have the following 8 steps clearly written and well organized:
STEP O: Read the problem and determine if you are testing hypotheses regarding a population proportion (p) or a population mean Obviously there is nothing to write for this step, but rather it is mean to help frame the problem.
STEP 1: GET PERMISSION!
For population means:
• Is the sample obtained by simple random sampling or the data a result from a randomized experiment?
e Is n > 30 (i.e., large)?
o If yes, we know from Central Limit Theorem that sampling distribution follows normal distribution, therefore no restrictions on the population distribution. o If no, (i.e., n < 30), it must be given that the population follows the normal distribution before you can conduct a hypothesis test.
For population proportions:
o Is the sample obtained by simple random sampling or the data a result from a randomized experiment?
o Is np>5 and n(l— p) 25 ?
STEP 2: SET UP THE HYPOTHESIS TEST.
Identify the null and alternative hypothesis statements (Ho and HI). Does the problem seek information regarding population mean (sigma known or unknown) or population proportion? Determine if you have a one-tailed test or two-tailed test? Which statement represents "status quo"? Which statement represents the claim being tested? Decide whether you are using a Classical Method Approach or a p-Value Approach.
STEP 3: REPRESENT a.—LEVEL, & GIVEN VALUES OF THE PROBLEM USING SYMBOLS. Identify values such as a, n, df = n — 1, G, s, x-bar, p-bar, critical values (z or t) or any other pertinent variable. (HINT: Not every problem will use all these values).
STEP 4: DRAW A PICTURE
Draw the normal distribution, with appropriate tail(s) shaded, label the alpha level(s), the "reject" and "do not reject regions," and any critical values (fence marks). Label and z-scale, t-scale, and/or x-scales, as appropriate.


STEP 5: CALCULATE TEST-STATISTIC (CLASSICAL) OR P-VALUE (the probability of observing a sample mean (or proportion) at least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is true). The p-value can also be described as the confidence you have that your null hypothesis is correct.
STEP 6: COMPARE TEST-STATISTIC WITH THE CRITICAL VALUE
STEP 7: MAKE YOUR DECISION-UDO NOT REJECT Ho" or "REJECT Ho"
Where does the test-statistic fall on the picture (if using Classical Approach)?
What is the relationship between the p-value and a-level (if using p-Value Approach)?   Explain why you made this decision, that is, what is your decision based upon?
STEP 8: WRITE A CONCLUSION STATEMENT USING ENGLISH WORDS IN A WELL FORMED SENTENCE (OR Two).
Based upon our sample Of sufficient evidence at the n "Is OR is not"
level of significance to conclude that ++++blah, blah, blah +++++ (use words as framed by the question).

Answer 1

STEP O:

We are testing hypotheses regarding a population mean.

STEP 1:

For population means:
• Is the sample obtained by simple random sampling or the data a result from a randomized experiment?

Yes
e Is n > 30 (i.e., large)?

Yes
o If yes, we know from Central Limit Theorem that sampling distribution follows normal distribution, therefore no restrictions on the population distribution. o If no, (i.e., n < 30), it must be given that the population follows the normal distribution before you can conduct a hypothesis test.

STEP 2:

The hypothesis being tested is:

H₀: µ = 1100  

H_a: µ ≠ 1100

This is a two-tailed test.

STEP 3:

x = 1080

µ = 1100

s = 80

n = 35

STEP 4:

The graph is:

STEP 5:

The test statistic, t = (x - µ)/s/√n = (1080-1100)/80/√35 = -1.479

The p-value is 0.1483.

The critical value is 2.032.

STEP 6:

1.479 < 2.032

STEP 7:

Since 1.479 < 2.032, we cannot reject the null hypothesis.

STEP 8:

Therefore, we have sufficient evidence to conclude that the mean lifetime of its lithium batteries is 1100 hours.