Question

1. Cough-a-Lot children’s cough syrup is supposed to contain 6 ounces of medicine per bottle. However, since the filling machine is not always precise, there can be variation from bottle to bottle. The amounts in the bottles are normally distributed with σ = 0.3 ounces. A quality assurance inspector measures 10 bottles and finds the following (in ounces):

   5.95

   6.10

   5.98

   6.01

   6.25

   5.85

   5.91

   6.05

   5.88

   5.91

   Are the results enough evidence to conclude that the bottles are not filled adequately at the labeled amount of 6 ounces per bottle?
   1. State the hypothesis you will test.
   2. Calculate the test statistic.
   3. Find the P-value.
   4. What is the conclusion?
2. Calculate a Z score when X = 20, μ = 17, and σ = 3.4.
3. Using a standard normal probabilities table, interpret the results for the Z score in Problem 12.
4. Your babysitter claims that she is underpaid given the current market. Her hourly wage is $12 per hour. You do some research and discover that the average wage in your area is $14 per hour with a standard deviation of 1.9. Calculate the Z score and use the table to find the standard normal probability. Based on your findings, should you give her a raise? Explain your reasoning as to why or why not.
5. O-Rama claims that their services will raise student SAT math scores at least 50 points. The average score on the math portion of the SAT is μ=350 and σ=35. The 100 students who completed the tutoring program had an average score of 385 points.
   1. Is the students’ average score of 385 points significant at the 5% and 1% levels to support O-Rama’s claim of at least a 50-point increase in the SAT score?
   2. Is the O-Rama students’ average score of 385 points significantly different at the 5% and 1% levels from the average score of 350 points on the math portion of the SAT? What conclusion can you make, based on your results, about the effectiveness of O-Rama’s?

Answer 1

Given :

Sample mean from the data given

Null and Alternative hypothesis :

H_o = Mean volume is equal to 6 oz i.e = 6

H_a = Mean Volume of bottle is not 6 oz i.e. 6

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Test Statistics

The z-statistic is computed as follows:

P-value

Corresponding to Z , the p-value is p=0.9077

Conclusion

Since p=0.9077≥0.05, it is concluded that the null hypothesis is not rejected. Therefore, we do not have sufficient evidence to conclude that the bottles are not filled adequately at the labeled amount of 6 ounces per bottle.

Z score when X = 20, μ = 17, and σ = 3.4