We continue the problem from \"In Class Work 5\". After studying 65 patients, we

Question

We continue the problem from "In Class Work 5". After studying 65 patients, we observed that our new watch reduced Parkinson's related tremors by 3.5 percentage points (x = 0.035) more than the old watch with a sample standard deviation of s^2 = 0.0076. We only wish to release the new watch if we are confident that it is at least 2 percentage points (0.02) better than the old watch at reducing tremors. (a) Assume our company wishes to operate at a 5% level of significance. Determine whether we should sell the watch. For full points include the null hypothesis, the alternative hypothesis, critical value(s)q, your test statistic and p-value, and explain the conclusion of your hypothesis test to a non-statistician. (b) Assume our company instead wishes to operate at a 5% level of significance. Determine whether we should sell the watch. For full points I only require the explanation given to a non-statistician. (c) Assume our company instead wishes to operate at a 1% level of significance. Determine whether we should sell the watch. For full points I only require the explanation given to a non-statistician.

Explanation / Answer

Solution

Note:

Only Part (a) is done in detail. For other parts, only critical values change and hence only the critical value and the final conclusion are given.

Part (a)

Let X = reduction (in percentage points) in tremors.

Then, X ~ N(µ, 2) where 2 is unknown.

Claim: The new watch meets the minimum stipulation on reduction in tremors.

Hypotheses:

Null H0: µ = µ0 = 0.02 Vs

Alternative HA: µ > 0.02

Test statistic:

t = (n)(Xbar - µ0)/s, where

Xbar = sample mean (given) = 0.035

µ0 (given) = 0.02

s = sample standard deviation = 0.0076

n = sample size = 65 (given)

So, tcal = (65)(0.035 – 0.02)/0.0076 = (65) x 1.9737 = 8.0623 x 1.9737 = 15.9528

Distribution, Critical Value and p-value

Under H0, t ~ tn - 1

Critical value = upper % point of tn - 1.

Given = 0.15, tcrit = t64, 0.15 = 1.045

p-value = P(t64 > 15.9528) = 2.77E-24

Decision Criterion (Rejection Region)

Reject H0, if tcal > tcrit or p-value < .

Decision:

Since tcal > tcrit, H0 is rejected. Also confirmed by p-value < .

Conclusion:

There is sufficient evidence to support the claim that the mean reduction in tremors is at least by 2% points.

In non-statistical terms, the test implies that it is unlikely that the reduction in tremors with the new watch is less than 2 percent points if the reduction in the sample is 3.5 percent points.

Part (b)

Critical value = upper 5% point of t64 = 1.669; p -value = P(t64 > 15.9528) = 2.77E-24

Decision:

Since tcal > tcrit, H0 is rejected. Also confirmed by p-value < .

Conclusion:

There is sufficient evidence to support the claim that the mean reduction in tremors is at least by 2% points.

Part (c)

Critical value = upper 1% point of t64 = 2.386; p-value = P(t64 > 15.9528) = 2.77E-24

Decision:

Since tcal > tcrit, H0 is rejected. Also confirmed by p-value < .

Conclusion:

There is sufficient evidence to support the claim that the mean reduction in tremors is at least by 2% points.

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