Problem

MATH1127: Introduction to Statistics (60022)
Lesson 9.3 Distribution Needed for Hypothesis Testing
Unit 3 Chapter 9: Lesso
Unit 3 Chapter 9: Lesson 9.3 Assignment
Score: $28 / 100$
$4 / 14$ answered
Question 13
A well-known brokerage firm executive claimed that at least $64 \%$ of investors are currently confident of meeting their investment goals. An XYZ Investor Optimism Survey, conducted over a two week period, found that out of 67 randomly selected people, 46 of them said they are confident of meeting their goals.
Suppose you are have the following null and alternative hypotheses for a test you are running:
\[
\begin{array}{l}
H_{0}: p=0.64 \\
H_{a}: p> 0.64
\end{array}
\]
Calculate the test statistic, rounded to 3 decimal places
\[
z=
\]
Question Help: D Post to forum
Submit Question

Answer

Expert–verified
Hide Steps
Answer

Rounding to 3 decimal places, the test statistic is \(\boxed{0.794}\).

Steps

Step 1 :The problem provides the following hypotheses for a test: \(H_{0}: p=0.64\) and \(H_{a}: p>0.64\). It also provides the sample size (n=67) and the number of positive responses (46).

Step 2 :We first calculate the sample proportion (\(\hat{p}\)) which is the ratio of the number of positive responses to the sample size. So, \(\hat{p} = \frac{46}{67} = 0.6865671641791045\).

Step 3 :The hypothesized population proportion (\(p_0\)) is given as 0.64.

Step 4 :We can now calculate the test statistic (z) using the formula: \[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]

Step 5 :Substituting the values into the formula, we get: \[z = \frac{0.6865671641791045 - 0.64}{\sqrt{\frac{0.64(1-0.64)}{67}}} = 0.7941013883159834\]

Step 6 :Rounding to 3 decimal places, the test statistic is \(\boxed{0.794}\).

link_gpt