Analysis Of Survey Results On Preference For A Unisex Gym - Essay.

Overview : To get marks my assignment has to show I am capable producing a report like the sample report.The following 7 sections demonstrate I can produce a report like the sample report 

Section 1 :Demonstrates I can reproduce the following computer output of the report using the sample allocated to me on moodle. Details of exactly how the computer does simple calculations is provided

1. Summary of the variable ‘does the customer want a unisex gym’ just considering the females
2. Summary that lets you investigate the relationship between the variable ‘time on cardio machine’ and the variable ‘time on weight machine’
3. Summary that lets you investigate the relationship between the variable ‘does the customer want a unisex gym’ and ‘gender’
4. Summary that lets you investigate the relationship between the variable ‘time spent on the cardio machine’ and ‘gender’

Survey Design and Data Collection

Australia there are many women-only gyms and since Gym Business XYZ has been successful enough to expand into a small new premises, customers were given a survey to determine if the new gym should be a unisex gym. 
A sample of 1000 customers were participated in the survey The following questions were posted to the respondents during the survey from which various responses were obtained.

Survey question 1: Are you male or female?
This question should be thought of a is a categorical variable, because the answer is one of the categories “male” or “female” Survey question 2: Would you prefer if the gym was unisex in other words if you are man would you prefer a gym with only men, yes or no?If you are woman would you prefer a gym with only women, yes or no?This question should be thought of a is a categorical variable, because the answer is one of 2 categories Survey question 3: How much time do you spend on the cardio machine (in minutes)?This question should be thought of as a Quantitative variable because it records the quantity (amount) of time spent on the machine .

Survey question 4: How much time do you spend on the weight machine (in minutes)? 
This question should be thought of as a Quantitative variable because it records the quantity (amount) of time spent on the weight machine The answers to the survey are available from appendix 1 

Methods of Data Analysis 
The data analysis presented in this report employs the following approaches;i.Simple descriptive statistics: Descriptive statistics are numbers and graphs that summarize sample. In our case the descriptive statistics are the sample mean, proportion, slope and intercept. The graphs are bar charts to compare proportions and a scatterplot to show the relationship between two quantitative variables.    

ii. Confidence intervals: Confidence intervals are found for the proportion parameters on the data. This was applied it testing the proportion of the respondents that wants unisex gym.
iii.Hypothesis tests: Hypothesis testing was conducted to determine whether there are relationships between among different variables such as gender, reasons for going to the gym, time on weight machine and time on cardio machine. 
Main Findings I.Females are more likely to prefer a unisex gym 
From the summary of the responses from the survey, the question as to whether females or males prefer unisex gym was answered positively by most female. The summary is as shown in the tables below.

    Should the Gym be Unisex    
Row Labels    No    Yes    Grand Total
Female    195    405    600
Male    270    130    400
Grand Total    465     535    1000

Count of Should the Gym be Unisex?    Column Labels        
Row Labels    No    Yes    Grand Total
Female    32.5%    67.50%    100%
Male    67.5%    32.5%    100%
Grand Total    46.5%    53.5%    100%

67.50% of the females want the gym to be unisex, however only 32.5% of males want a unisex gym, so 7.50%-32.5%=35.0 % more women want the gym to be Unisex 

Methods of Data Analysis

ii. Females spend more time on the Cardio machine

Based on this research question, it is necessary to test whether females spend much time on the cardio machine or not. The average mean of minutes on the cardio machine and the standard deviation of the minute spent on the cardio machine were analysed for both the gender
Row Labels    Count of gender    Average of Minutes on Cardio    Std Dev of Minutes on Cardio
Female    600    39.58    17.351
Male    400    14.625    13.155
Grand Total    1000    28    19.14854216

From the table above, it is crystal clear that females spent an average of 38 minutes on the cardio machine while males spent an average of 14 minutes. Thus, females spend 24 more minutes on average.

iii. The time spent on cardio machine increases with a decrease in the time spend on the weight machine

This research question aimed at predicting whether there exist an association between time spent on the cardio machine and the time spent on the weight machine.
It is evident from the regression analysis that as the time spent on the cardio machine increases the time spent on the weight decreases. The scatter plot shows the relationship between the variables.

Confidence Intervals

Confidence intervals for the proportion of females that want a unisex gym and the Proportion of males that want a unisex gym.i.The 90% confidence interval for sample proportion for females The tables below shows the output of the results for sample proportions of females and males

    Should the gym by Unisex    
Row Labels    No    yes    Grand Total
Female    21    39    60
Male    32    8    400
Grand Total    53    47    1000
 

By considering sample number of females n=600, sample proportion = 405/600=0.675 and standard error of sample proportion =  0.0715, 90% confidence interval for the proportion becomes; 0.65< Proportion<0.795.i.The 90% confidence interval for sample proportion for males 

    Should the Gym be Unisex    
Row Labels    No    yes    Grand Total
Female    195    405    600
Male    270    130    400
Grand Total    465    535    1000
    
By taking sample for males n=400, sample proportion = 270/400=0.675 and standard error of sample proportion =  0.0676 the 90% confidence interval for the true proportion becomes; 0.095<P<0.305

Hypothesis tests that Support the Main Findings  i.Female
The hypotheses for the survey were formulated as follows;
Null hypothesis (H0): The gym should be unisex i.e. ≤0.5 (to find the test stat assume p=0.5)
Alternative hypothesis (HA): The gym should not be unisex i.e. p>0.5
Then the test statistics becomes;   

    Should the Gym be unisex    
Row Labels    No    yes    Grand Total
Female    195    405    600
Male    270    130    400
Grand Total    465    535    1000 

=

The p-value is P (Z>1.324) =0.01

Since the p-value is less than 0.05 we accept H0 because there is strong evidence the proportion is above 50%, and thus we conclude that most females need a unisex gym.ii.Males 

    Should the Gym be Unisex    
Row Labels    No    yes    Grand Total
Female    195    405    600
Male    270    130    400
Grand Total    465    535    1000
     
Buy considering the males n=40, sample proportion =8/40=0.2 and test statistics becomes=

The p-value is P (Z>-3.7947) =0.999926 and since the p-value is less than 0.05 we do reject H0 and thus we conclude that there is no strong evidence to suggest that most of the males want a unisex gym. Hypothesis Testing Comparing Proportions  Before giving the survey to the customers, it is was clear that the summary of the dataset can be used to answer to the question “Is there a relationship between variables ‘gender’ and ‘should the gym be unisex’ “
So the appropriate null and alternative hypothesis is  H0 :p1= p2  HA :p1≠ p2
the appropriate test stat formula is
 Where  since we assume Ho to true this becomes 
Using the information from the sample, the p value becomes; = 4.41
P-value is  P (Z<-4.66)+P(Z>5.31)=0.000

Hypothesis test comparing means

Here hypothesis testing was done to determine whether there exist a relationship between gander and time on cardio machine. The hypotheses were formulated as follows;
H0 :µ1= µ2  HA :µ1≠ µ2
The appropriate test statistic is 
  Which has a t distribution with df 
By assuming H0 to true to calculate the test stat, the test stat becomes 

Results/Findings 
Row Labels    Count of gender    Average of Minutes on Cardio    Std Dev of Minutes on Cardio
Female    600    37.5833    16.35
Male    400    13.62    13.15
Grand Total    1000      28    19.14
=94   test stat =7.883
P-value is P (t<-8.08) +P (t>8.08) = 0.000

P-value 0.000
Thus we reject null hypothesis and conclude that there is a difference between the means of the variables. 
Answer: 
The test stat is 

Since you assume the null hypothesis is true the test stat becomes
Test stat=
The formulas for  and  are very complicated.
                
     Coefficients    Standard Error    t Stat    P-value
Intercept    34.48099174
Minutes on Cardio    -0.702892562    0.034814    -20.1    1.07E-36

      Test stat=
P-value is P (t<-20.1) +P (t>20.1)=1.07E-36
Considering the above points, women’s gyms are definitely more suitable for women who want to get into shape or just stay fit. They provide the ideal environment for safe and peaceful workouts.

Based on the analysis, there are strong evidences which suggest that most females want a Unisex gym and they concentrate on buying cardio machine. Most males are happy with a mixed gender gym. So if Gym business XYZ buys a small new premises then it should make the gym only for females.