Question

A sample of 14 small bags of the same brand of candies was selected. Assume that the population distribution of bag weights is normal. The weight of each bag was then recorded. The mean weight was 3 ounces with a standard deviation of 0.15 ounces. The population standard deviation is known to be 0.1 ounce.

NOTE: If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

• PART (a)

  Find the following. (Round your answers to two decimal places.)
• (i)   x = _____
  (ii)   σ = _____
  (iii)   s_x = _____

• PART (b)

  In words, define the random variable X.

  A) the mean weight of the candies

• B) the number of candies in a bag    

• C) the weight, in ounces, of a piece of candy

• D) the weight, in ounces, of one bag of candy

• PART (c)

• In words, define the random variable X.

  A) the mean weight, in ounces, of the pieces of candy

• B) the mean number of candies from each bag that weigh 3 ounces    

• C) the mean number of candies for the sample of 14 bags

• D) the mean weight, in ounces, for the sample of 14 bags

• PART (d)

  Which distribution should you use for this problem? (Round your answers to three decimal places.)

  X ~ ____ (____,____)

• Explain your choice

• A) The Student's t-distribution should be used because the sample standard deviation is given.

• B) The Student's t-distribution should be used because the sample size is small.    

• C) The standard normal distribution should be used because the sample standard deviation is known.

• D) The standard normal distribution should be used because the population standard deviation is known.

• PART (e)

  Construct a 90% confidence interval for the population mean weight of the candies.
• (i) State the confidence interval. (Round your answers to three decimal places.)
• (_____,_____)
• (ii) Sketch the graph.
• _________ _________
• (iii) Calculate the error bound. (Round your answer to three decimal places.)
• _______

• PART (f)

  Construct a 98% confidence interval for the population mean weight of the candies.
• (i) State the confidence interval. (Round your answers to three decimal places.)
• (_____,_____)
• (ii) Sketch the graph.

  
  
  

  _______ _______
• (iii) Calculate the error bound. (Round your answer to three decimal places.)
• ________

• PART (g)

  In complete sentences, explain why the confidence interval in part (f) is larger than the confidence interval in part (e).

  A) The confidence interval in part (f) is larger than the confidence interval in part (e) because a small sample size is being used.

• B) The confidence interval in part (f) is larger than the confidence interval in part (e) because the population standard deviation changes for each sample.   

• C) The confidence interval in part (f) is larger than the confidence interval in part (e) because a larger level of confidence increases the error bound, making the interval larger.

• D) The confidence interval in part (f) is larger than the confidence interval in part (e) because the mean weight changes for each sample.

• PART (h)

  In complete sentences, give an interpretation of what the interval in part (f) means.

  A) We are 98% confident that the mean weight of the sample of 14 small bags of candies is between these values.

• B) We are 98% confident that a small bag of candies weighs between these values.    

• C) There is a 98% chance that a small bag of candies weighs between these values.

• D) We are 98% confident that the true population mean weight of all small bags of candies is between these values.

Answer 1

a ) i )

ii )

iii )

b ) D )  the weight, in ounces, of one bag of candy

c ) D) the mean weight, in ounces, for the sample of 14 bags

d )

D) The standard normal distribution should be used because the population standard deviation is known.