A sample of 36 resistors taken from a production line gave a mean resistance of

Question

A sample of 36 resistors taken from a production line gave a mean resistance of 54.089 omega, with a Sample standard deviation of 4.85 omega The production line should produce resistors with an overall mean resistance of 50.02 omega. Test at the 5' level of significance that whether the mean resistance of the production output is as it should be? Answer the below questions for this hypothesis test: H_0 = Number H_1 notequalto Number Do we use Z-test? Yes no What are the critical values: Left tail critical value = Number right tail critical value = Number Evaluate the Z or t value and enter your answer rounded correct to 3 decimal places: Number Reject the null hypothesis yes no

Explanation / Answer

Solution

Let X = Resistance in ohms of the resistors

We assume X has a Normal Distribution with mean µ and variance ^2. Neither µ nor is known.

Answering the question is equivalent to testing if µ = 50.02 when ^2 is unknown.

Null Hypothesis H0: µ = 50.02 vs Alternative H1: µ 50.02

Level Significance: 5% (given)

Test Statistics: t = (n)(Mean X - 50.02)/s, where Mean X = sample average, s = sample standard deviation and n = sample size. Under H0, t has a t-distribution with degrees of freedom = n - 1.

From the given data, Mean X = 54.089, s = 4.85, n = 36

Calculations: t = 6(54.089 - 50.02)4.85 = (6 x 4.039)/4.85 = 24.234/4.85 = 4.997

Decision Criterion: Reject  H0 if calculated value of t > critical value (in this case upper 2.5% point of t distribution with degrees of freedom = 35, which is found to be 2.031 from Standard Table)

Since calculated value of t > critical value, H0 rejected

Conclusion: Mean resistance of the production is not as per requirement.

[Additional inputs:

1. In this case we employed t-distribution since ^2 is unknown. If it were known, we would have used Z-test.

2. This is a two-sided test since both more than and less than the required mean of 50.02 are not acceptable.

3. Since the test is two-sided, critical value corresponds to half the level of significance.]

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