Question

You will be performing an analysis on female heights, given of set of 30 heights that were
randomly obtained. For this project, it is necessary to know that the average height for women
is assumed to be 65 inches with a standard deviation of 3.5 inches. You will use these numbers
in some of your calculations.

Height (in Inches)	Name
72.44	Emma
67.53	Olivia
66.71	Ava
62.02	Isabella
73.89	Sophia
65.95	Mia
65.83	Charlotte
64.15	Amelia
65.39	Evelyn
59.68	Abigail
64.24	Harper
66.60	Emily
65.40	Elizabeth
64.72	Avery
67.11	Sofia
61.97	Ella
62.83	Madison
67.20	Scarlett
66.62	Victoria
68.78	Aria
66.13	Grace
64.47	Chloe
66.64	Camila
62.39	Penelope
63.90	Riley
62.97	Layla
59.31	Lillian
66.14	Nora
67.54	Zoey
63.45	Mila

1. Find Elizabeth in the data and, given the population mean and standard deviation,
calculate:
a. The z-score for Elizabeth (easiest to be done by hand).
b. The probability that a randomly selected female is shorter than Elizabeth.
c. The probability that a randomly selected female is taller than Elizabeth.
d. Interpret each of these in a sentence (3 full sentences).

2. As you know, a random sample of 30 women’s heights was obtained. Describe the
sampling distribution. Use a full sentence here to describe the sampling distribution.

3. Find and state the mean of the 30 women’s heights that were
provided (round to two decimal places). Then (using the values from step 2), calculate
the probability that a random sample of 30 women’s heights would result in a mean
that was the value you found or more. Use this probability in a full sentence.

4. Explain how what you just calculated in step 3 relates to the question you were asked in
step 1c. Why is the probability in step 3 lower than that in step 1?

5. For this step, please work under the assumption that we do not know the population
mean and standard deviation (and in fact, if we are running this test, it means that we
are not sure of these values). Construct a 95% confidence interval for the average height
of a female. Interpret this confidence interval.

6. It is a researcher’s belief that the average height of females has increased since the
average value of 65 inches was reported. Using the sample of 30 female heights,
conduct a hypothesis test at the .05 level that tests whether or not this researcher might
be correct. You must state the null and alternative hypotheses as well as the p-value and
an interpretation of what this means.

7. How do the results from steps 5 and 6 relate to one another?

I know I have many questions... but I need your help. Please help me with those questions and I will upvote the answer.

Answer 1

1.(a) Here, Mean is given by = 65 inches

Standard Deviation = 3.5 inches

And Z score =

Height of Elizabeth = x = 65.4 inches

Thus, Z = (65.4-65) / 3.5

= 0.4 / 3.5

= 0.1142

(b)Value Of Z = 0.1142 in normal distribution is 0.3965

Therefore, Probability that a randomly selected girl is shorter than Elizabeth is given by

and

So, we need to find

= 0.5 - 0.3965

= 0.1035

Since we need to find

So, = 1 - 0.1035

= 0.8965

(c) Probability that a randomly selected girl is taller than Elizabeth is given by

= 0.5 - 0.3965

= 0.1035

(d) The interpretation of the above three has been done along with the solution.