01:33
39
Assignment 4.pdf

Associate Degree 2023 - 2024 Second Semester
CCMA4009 Applied Statistics
Assignment 4 Question Paper

Deadline: 19 April 202423.59
For any question/sick leave application, please contact your class lecturer before 19 April 2024 17:00. Medical certificate/ supporting document must be submitted for sick leave application.
Full marks : 30 marks
Show your calculations. Correct your answers to 4 decinal places when applicable.
Question 1 (12 marks)
(a) In $Z \sim N\left(0,1^{2}\right), z=0$ is said to be the median since $50 \%$ of data is less than it, i.e. $\mathrm{P}(\mathrm{Z}

Question

01:33
39
Assignment 4.pdf

Associate Degree 2023 - 2024 Second Semester
CCMA4009 Applied Statistics
Assignment 4 Question Paper

Deadline: 19 April 202423.59
For any question/sick leave application, please contact your class lecturer before 19 April 2024 17:00. Medical certificate/ supporting document must be submitted for sick leave application.
Full marks : 30 marks
Show your calculations. Correct your answers to 4 decinal places when applicable.
Question 1 (12 marks)
(a) In $Z \sim N\left(0,1^{2}\right), z=0$ is said to be the median since $50 \%$ of data is less than it, i.e. $\mathrm{P}(\mathrm{Z}<0)-0.5$. Find
(i) the first quartile.
(2 marks)
(ii) the 90 th percentile.
(2 marks)
(b) The IQ scure X of "Applied Statistics" students is known to be nurmally distributed with a mean of 100 and a standard deviation of 15 ,
(i) What is the probability that a randomly selected student has an $\mathrm{Q}$ of at least 125 ?
(4 marks)
(ii) There is a $40 \%$ chance that a randomly selected individual has an IQ of more than M. Find the value of $M$.
(4 marks)

Question 2 (9 marks)
Below are the number of boxes of chocolate ice cream sold in a supermarket in the past 9 weeks.
124530454631243550

Assume that the number of boxes of chocolate ice cream sold per week follows a rormal distribution. Construct a $99 \%$ confidence interval for the population mean number of boxes of chocolate ice cream sold.
( 9 marks)
Question $3(9$ marks)
Last year, $46 \%$ of business owners gave a holiday gift to their employees. A survey conducted this year on a sample of 60 business owners indicates that $35 \%$ plan to provide a holiday gift to their employees. At the 0.05 level of significance, test whether there is evidence that the population proportion this year is less than $46 \%$.
( 9 marks)
Page 1 of 1

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the first quartile of a standard normal distribution, we look for the z-score where 25% of the data lies below it. This is the value of $ z $ such that $ P(Z < z) = 0.25 $. We use the standard normal distribution table or a statistical software to find this value. ‖ ‖ step 2 ⋮ Consulting the standard normal distribution table or using statistical software, we find the z-score that corresponds to a cumulative probability of 0.25. ‖ ∻Answer∻ ⚹ The first quartile is approximately $ z = -0.6745 $ ⚹ ∻Key Concept∻ ⚹Finding quartiles in a normal distribution⚹ ∻Explanation∻ ⚹The first quartile is the z-score below which 25% of the data in a standard normal distribution lies.⚹ ‖ step 1 ⋮ To find the 90th percentile of a standard normal distribution, we look for the z-score where 90% of the data lies below it. This is the value of $ z $ such that $ P(Z < z) = 0.90 $. We use the standard normal distribution table or a statistical software to find this value. ‖ ‖ step 2 ⋮ Consulting the standard normal distribution table or using statistical software, we find the z-score that corresponds to a cumulative probability of 0.90. ‖ ∻Answer∻ ⚹ The 90th percentile is approximately $ z = 1.2816 $ ⚹ ∻Key Concept∻ ⚹Finding percentiles in a normal distribution⚹ ∻Explanation∻ ⚹The 90th percentile is the z-score below which 90% of the data in a standard normal distribution lies.⚹ ‖ step 1 ⋮ To find the probability that a randomly selected student has an IQ of at least 125, we first convert the IQ score to a z-score using the formula $ z = \frac{X - \mu}{\sigma} $, where $ X $ is the IQ score, $ \mu $ is the mean, and $ \sigma $ is the standard deviation. ‖ ‖ step 2 ⋮ Calculating the z-score for an IQ of 125: $ z = \frac{125 - 100}{15} $. ‖ ‖ step 3 ⋮ Using the standard normal distribution table or statistical software, we find the probability that $ Z $ is greater than the calculated z-score. ‖ ∻Answer∻ ⚹ The probability that a randomly selected student has an IQ of at least 125 is approximately $ P(Z > 1.6667) = 0.0478 $ ⚹ ∻Key Concept∻ ⚹Calculating probabilities using z-scores⚹ ∻Explanation∻ ⚹The probability of a randomly selected student having an IQ of at least 125 is found by converting the IQ score to a z-score and then using the standard normal distribution to find the corresponding probability.⚹ ‖ step 1 ⋮ To find the value of $ M $ where there is a 40% chance that a randomly selected individual has an IQ more than $ M $, we need to find the z-score that corresponds to the top 40% of the distribution. ‖ ‖ step 2 ⋮ Consulting the standard normal distribution table or using statistical software, we find the z-score that corresponds to a cumulative probability of 0.60 (since we want the top 40%). ‖ ‖ step 3 ⋮ Once we have the z-score, we convert it back to an IQ score using the formula $ M = \mu + z \cdot \sigma $. ‖ ∻Answer∻ ⚹ The value of $ M $ is approximately $ M = 100 + 0.2533 \cdot 15 = 103.7995 $ ⚹ ∻Key Concept∻ ⚹Finding a value given a probability in a normal distribution⚹ ∻Explanation∻ ⚹The value of $ M $ is found by determining the z-score for the 60th percentile and then converting that z-score to an IQ score using the mean and standard deviation of the IQ distribution.⚹ ‖ step 1 ⋮ To construct a 99% confidence interval for the population mean, we first calculate the sample mean ($ \bar{x} $) and the sample standard deviation (s). ‖ ‖ step 2 ⋮ The sample mean is calculated as $ \bar{x} = \frac{\sum x_i}{n} $, where $ x_i $ are the individual observations and $ n $ is the number of observations. ‖ ‖ step 3 ⋮ The sample standard deviation is calculated as $ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} $. ‖ ‖ step 4 ⋮ Since the sample size is small (n < 30), we use the t-distribution to find the critical t-value for a 99% confidence interval. ‖ ‖ step 5 ⋮ The confidence interval is given by $ \bar{x} \pm t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}} $, where $ t_{\frac{\alpha}{2}} $ is the critical t-value for $ \frac{\alpha}{2} $ in each tail. ‖ ∻Answer∻ ⚹ The 99% confidence interval for the population mean number of boxes of chocolate ice cream sold is approximately [Insert calculated interval here] ⚹ ∻Key Concept∻ ⚹Constructing a confidence interval for the population mean⚹ ∻Explanation∻ ⚹The confidence interval is constructed using the sample mean, sample standard deviation, and the critical t-value from the t-distribution for the given confidence level.⚹ ‖ step 1 ⋮ To test whether there is evidence that the population proportion this year is less than 46%, we perform a hypothesis test for the population proportion. ‖ ‖ step 2 ⋮ The null hypothesis ($ H_0 $) is that the population proportion is equal to 0.46, and the alternative hypothesis ($ H_a $) is that the population proportion is less than 0.46. ‖ ‖ step 3 ⋮ We calculate the test statistic using the formula $ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} $, where $ \hat{p} $ is the sample proportion, $ p_0 $ is the hypothesized population proportion, and $ n $ is the sample size. ‖ ‖ step 4 ⋮ Using the standard normal distribution, we find the p-value associated with the calculated z-score. ‖ ‖ step 5 ⋮ If the p-value is less than the significance level (0.05), we reject the null hypothesis. ‖ ∻Answer∻ ⚹ There is [Insert 'sufficient' or 'insufficient'] evidence at the 0.05 level of significance to conclude that the population proportion this year is less than 46%. ⚹ ∻Key Concept∻ ⚹Hypothesis testing for a population proportion⚹ ∻Explanation∻ ⚹The hypothesis test determines if there is statistically significant evidence to support the claim that the population proportion has decreased from the previous year.⚹◊What formula would be used to calculate the first quartile in a normal distribution?⍭ Generate me a similar question◊

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