Question

13. A company that sells thousands of bicycles online maintains a telephone help line to assist customers who are

assembling their bicycle. The company keeps detailed records of the percentage of customers who purchase
a bicycle who call in for help. The company claims that 18% of all customers who purchase a bicycle call the
help line.
a. Let p= the proportion of customers in a sample who call the help line. Calculate the mean and standard
deviation of the sampling distribution of p for random samples of size 100 from this population.
b. Interpret the standard deviation from part (a). SE 0.039 SD: 0.18
c. Would it be appropriate to use a normal distribution to model the sampling distribution of p? Justify your
answer.
d. In an effort to reduce the percentage of customers who call in for help, the assembly instructions were
rewritten with improved illustrations. The company selects an SRS of 100 purchasers who received the
instructions with the improved illustrations. What is the probability that at most 14% of the customers in
the sample call the help line?
e. The company finds that 14% of the customers in the sample called the help line. Based on your answer
to part (d), do these data provide convincing evidence that the proportion of customers who would call
the help line when given the improved assembly instructions is less than 0.18? Explain your reasoning.

Answer 1

The question discusses the application of statistical methods to determine the mean, standard deviation, and appropriate distribution model for a given proportion of customer calls to a helpline, and whether the improved instructions were effective by using a Z-test.

Step-by-step explanation:

The subject of this question falls under the category of Mathematics, more specifically, topics related to statistics and probability, which are often covered in high school curriculum.

a. To calculate the mean (μ_p) of the sampling distribution of proportion (p), we use the formula μ_p = P, where P is the population proportion. For the standard deviation (σ_p), we use the formula σ_p = sqrt(P(1 - P) / n). Given that P = 0.18 and n = 100, we have:

• Mean (μ_p) = 0.18
• Standard Deviation (σ_p) = sqrt(0.18 * 0.82 / 100) = 0.039

b. The standard deviation of the sampling distribution (0.039) measures the variability of sample proportions around the population proportion. It represents how much the proportion in a sample of size 100 is expected to differ from the population proportion of 0.18.

c. The use of normal distribution to model the sampling distribution of p is appropriate if the sample size is large enough and if np and n(1-p) are both greater than 5. Here, 100 * 0.18 = 18 and 100 * 0.82 = 82, which meets the criteria, so a normal distribution can be used.

d. To find the probability that at most 14% of the customers in the sample call the help line, we need to use the normal approximation of the binomial: Z = (p - P) / σ_p. For p = 0.14, Z = (0.14 - 0.18) / 0.039. We would then look up this Z-value in a standard normal table or use a calculator to find the corresponding probability.

e. If the sample shows that 14% of the customers called the help line, this is lower than the claimed 18%. Using the Z-value calculated in (d) and the normal distribution table or calculator, we can check if the result is statistically significant. If the probability of obtaining a sample proportion of 14% or less is low under the assumption that the true proportion is 18%, it provides evidence against the null hypothesis and supports the claim that the improved instructions reduced the percentage of calls for help.