19. A salesman is expected to effect an average sales of Rs. 3500. A sample test revealed that a particular sales man had made the following sales. Rs 3700, 2500, 3400, 5200, 3000 and 2000 Using .05 level of significance conclude whether his work is below
standard or not.
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Answers
Answer:
Birth Weights in a Town
Let’s return to the familiar context of birth weights for babies in a town. Suppose that babies in the town had a mean birth weight of 3,500 grams in 2010. This year, a random sample of 50 babies has a mean weight of about 3,400 grams with a standard deviation of about 500 grams. Here is the distribution of birth weights in the sample.
Dot plot of birth weights, ranging from around 2,000 grams to 4,000 grams.
Obviously, this sample weighs less on average than the population of babies in the town in 2010. A decrease in the town’s mean birth weight could indicate a decline in overall health of the town. But does this sample give strong evidence that the town’s mean birth weight is less than 3,500 grams this year?
We now know how to answer this question with a hypothesis test. Let’s use a significance level of 5%.
Let μ = mean birth weight in the town this year. The null hypothesis says there is “no change from 2010.”
H0: μ < 3,500
Ha: μ = 3,500
Since the sample is large, we can conduct the T-test (without worrying about the shape of the distribution of birth weights for individual babies.)
\displaystyle T\text{}=\text{}\frac{\mathrm{3,400}-\mathrm{3,500}}{\frac{500}{\sqrt{50}}}\text{}\approx \text{}-1.41T=
√
50
500
3,400−3,500
≈−1.41
Statistical software tells us the P-value is 0.082 = 8.2%. Since the P-value is greater than 0.05, we fail to reject the null hypothesis.
Our conclusion: This sample does not suggest that the mean birth weight this year is less than 3,500 grams (P-value = 0.082). The sample from this year has a mean of 3,400 grams, which is 100 grams lower than the mean in 2010. But this difference is not statistically significant. It can be explained by the chance fluctuation we expect to see in random sampling.