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Question 1

a) Mean, median, and mode for each quiz

Table 1: Calculation of Mean, median, and mode for each quiz

b) Measures of Centre or Measures of Central Tendency

Here, according to the above table, the median value is seen to be the most central and middlemost value. Hence, the central tendency is determined to be 72.

c) Weaknesses or Strengths of each statistic of centre

Strengths

There are three statistics of centre, which includes Mean, Median and Mode. The mean involves the participation of each data, thus it is representation of all the data. Most importantly, it a good representation as this value hardly appears in the raw data.

As in the 4 given cases, where there are total marks of four quizzes, the mean value appeared to be 72 which had not appeared in any of the data of each quiz. However, if it is drawn repeated samples from the same set of data, then, the mean tend to possess similar nearby values. Moreover, it is highly similar to the standard deviation, which is one of the common measures of dispersion.

Weaknesses

In case of non-nominal or nominal value, it is not possible to calculate the mean value. The main disadvantage of the Mode is that, if all the data appears only once in the raw data, then there will not be any modal value, so it is not possible to deduce the Mode.

d) Symmetric or Skewed data and the direction

Quiz 1

Table 2: Calculation of Skewed Data

Figure 1: Histogram showing Right Skewed Data (Positive Skewness)

Quiz 2

Table 3: Calculation of Skewed Data

Figure 2: Histogram showing Symmetric Data

Quiz 3

Table 4: Calculation of Skewed Data

Figure 3: Histogram showing Right Skewed Data (Positive Skewness)

Quiz 4

Table 5 Calculation of Skewed Data

Figure 4: Histogram showing Left Skewed Data (Negative Skewness)

e) Comparison and Description of the performance of each student

In each quiz, all the students had performed well. In quiz 1 and 2, the students got Right Skewed Data, while in quiz 4, they got Left Skewed Data in Quiz 3, and all data are symmetric.

Question 2:

a) Construction of a 90% confidence interval with respect to the true proportion

By considering the provided data, it has been seen that the sample size (n) = 100 and frequency (x) = 19, and P is the proportion = x/ n = 0.19

Confidence interval of the mean percent is 90% that is 0.9.

The formula for constructing a 90% confidence interval = P ± Z

Now, (ɑ/ 2) * √((P * Q)/ n) (x, n - x ≥ 5), and Q = 1 - P = 1 - 0.19 = 0.81.

ɑ = 1- (level of confidence/ 100) = 1 - 0.9 = 0.1.

Z * (ɑ/ 2) = Z - vale of the table.

(ɑ/ 2) * √((P * Q)/ n) = (0.1/ 2) * √((0.19 * 0.81)/ 100) = (0.1/ 2) * √(0.1539/ 100) = 0.05 * 0.001539 = 0.00007695 (Approximately).

1 - ((ɑ/ 2) * √((P * Q)/ n)) = 1 - 0.00007695 = 0.99992305.

Therefore, the interval is 0.001539 to 0.99992305.

b) Explanation on whether the normality is being assumed or not

In the data n = 100 and 𝝿 = 0.9

Therefore, the standard can be found as, √((𝝿 * (1 - 𝝿)/100) = √((0.9 * 0.1)/100) = 0.03

Hence, as the standard error is not of both the positive and negative sign, normality cannot be assumed. This is because for the existence of normality the values need to be distributed along with mean in the sample.

c) Finding the requirement of the sample size for 90% confidence and the error of ± 0.03

For finding out the sample size with respect to the 90% confidence and the error of ± 0.03, the following formula has been considered:

Figure: formula for sample size

In accordance with the 90% confidence interval, Z = 1.65, and in the data, it has been given that N = 100 and e = ± 0.03.

Therefore, sample size = ((1.65^2 * 0.19 * (1 - 0.19))/(0.03^2))/ (1 + (1.65^2 * 0.19 * (1 - 0.19))/(0.03^2 * 100))) = 465.5475/ 4.655475 = 100.

Therefore, the sample size is 100.

d) Reason behind the needs of understanding the sampling for a quality control manager

A quality control manager always seeks to understand the sampling in a specific manner. As per the statement of Mitra (2016, p 45), it can be said that the quality control manager needs to prepare the plan for the quality of the product to be delivered to the customers. The quality control manager also looks after the evaluation of the overall performance of the project on a daily basis. In accordance with the words of Okonechnikov et al. (2015, p. 293), it can be said that the monitoring procedures of the quality control manager will help them to regulate the confirmation on the quality of the delivered product for developing customer satisfaction. As per the words of Price (2017, p. 56), it can be said that with the help of supervision on the overall project the managers can easily remove the gaps between the expected quality and the delivered quality of the products or services. In the given case, it has been seen that the project is related to deliver mixed nuts. Therefore, to deliver nuts of better quality the quality control manager at Planter's may need to understand the sampling to minimise the errors in the project delivery.

Question 3:

a) Finding whether the price coefficient is different significantly from Zero at ɑ = 0.05

Population of correlation coefficient is being denoted by ⍴ and the sample of the correlation coefficient is being denoted by n. In accordance with the given data, it can be said that the ⍴, that is the population is unknown and the sample has been derived in the given result of the regression analysis. It has been given that quality is being expressed as price function and the sample observations are equal to 27, that is n = 27.

It has been also given that ɑ = 0.05.

Therefore, ɑ/ 2 = 0.05/2 = 0.025.

Furthermore, from the value of n = 27, n - 2 = 27 - 2 = 25.

In the given data, it has been given that t Stat for the respective price coefficient is = -0.528.

Hence, it can be concluded that the price coefficient of the given regression analysis is significantly different from Zero, with respect to ɑ = 0.05.

b) Representation of R^2

In order to find the way of fitting the data into a regression line, a statistical measure can be found in the process of setting the regression line. The respective measure of statistics helps to find the closeness among the data that have been fit into the regression line. This measures has been known as the R^2. as per the comments provided by Fox (2015, p. 56), it has been seen that this statistical measures of R^2 can also be used as a coefficient of determination as well as with respect to the multiple lines of regression the measure is considered as the coefficient for the multiple determination. In accordance with the statements provided by Price (2017, p. 89), it can be said that if the value of R^2 has been found as 0 or the coefficient of the determination or the regression is 0%, then it represents that the model of the linear regression is none.

c) Conclusion on the statement of higher prices refer to higher level of quality

According to the price as well as the quality equilibrium, it can be said that the statement of higher prices of higher level of quality is true in some cases. It can also be said that this statement is not always true. As per the words of Achmad et al. (2015, p. 241), it can be said that the respective statement can be true in the concerned case of stereo speakers. This is because this case is related to technologies, which has been measured according to the value of prices. Therefore, in the respective case it can be said that higher prices for the stereo speakers there will have more advanced specifications in the products. However, after a certain amount of price for the products will have the need to be increased, after reaching the higher level of quality for the respective products.

Question 4:

a) Setting up of Null as well as alternative hypothesis

Considering the statements of statistics related to the hypothesis, the null hypothesis refers to the hypothesis, which reflects that among the statistical variables there will be no statistical significance related to the operation of the respective business. In accordance with the statements of Nathoo and Masson (2016, p. 149), it can be said that the null hypothesis has been considered at the time of testing disprove of the assumption. In the respective case of the pick-up and delivery business, if the picking up process is being related to the production process of the organisation, it will be a null hypothesis. As per the words of Gallagher et al. (2017, p. 399), the alternative hypothesis has been known as the state, where the statistical values have been related in a statistical significance.

b) Identification of the type of error due to wrong conclusion about delivery within two days and impacts of this error

According to the respective business of pickup and delivery, if the delivery has been assumed to be done within two days the error has been known as type I error. This represents that quality may be lower as compared to on-time delivery, as this refers to before time delivery.

c) Identification of the type of error due to wrong conclusion about delivery at more than two days and the relative impacts

According to the respective business of pickup and delivery, if the delivery has been assumed to be done more than two days the error has been known as type II error. This refers that the delivery will exceed the delivery period. This represents that the company may run in loss as compared to on-time delivery.

d) Finding out the worse error from the standpoint of the company with reason

Type II delivery is known as the worse error for the company. This is because type II error leads the process of the business in loss.

e) Explanation of the worse error from the standpoint of customer with reason

Type I delivery is known as the worse error for the customer. This will represent the delivery before the time of delivery, which may lead to get lower level in the quality.