Why theGym Business XYZshould start aseparate womenonly gym after it hadsufficient growth. Executive summary GymBusiness XYZis planning to expandbypurchasingan extra smaller nearbylocation,based on...

Why theGym Business XYZshould start aseparate womenonly gym after it hadsufficient growth.

Executive summary

GymBusiness XYZis planning to expandbypurchasingan extra smaller nearbylocation,based on this reportit should makethenew smallgym a Unisex Gymthat will only serve females.

Contents

Background and data set​2

Methods used to analyse the survey data​3

Main findings​4

Confidence intervals​6

Hypothesis tests that support the main findings​7

Conclusion​16

Appendix part 1 data set​17

Appendix part 2 notes on hypothesis testing​20

Appendix 3 regression formula​24

Backgroundand data set

In Australia there are many women only gyms and since Gym Business XYZ has been successful enough to expand into asmall new premises customerswere given a survey to determine if the new gym should be a unisex gym.

A sample of 100 customers were given the following survey

Survey question 1:Are you male or female?

This question should be thought of a is a categorical variable,becausethe answer is one of the categories “male” or “female”

Survey question2: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 of2categories

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 usedto analyse the survey data

This report will use

*Simple descriptive statistics

*Confidence intervals

*Hypothesis tests

Going into more detail

Simple descriptivestatistics:Descriptive statistics are numbers and graphs that summarizesample. 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.

Confidence intervals:

Confidence intervals are found for the following parameters related to the report

*Just considering the females, what proportion of females want a unisex gym.
*Just considering the males, what proportion of males want a unisex gym.

Hypothesistests:

The main findings of this report are related to the following Hypotheses

*Do most women want a Unisex gym?

*Do Most men want a Unisex gym?

*Is there a relationship between the variables “Gender” and “Reason for going to the gym”

*Is there a relationship between the variables “Time on weight machine” and “Time on cardio machine”

This report usesHypothesis tests to check ifthere is strong evidence for a hypothesis testRefer toappendix2for notes on hypothesis testing

Main findings

Females are more likely to prefer a unisex gym

This is obvious if you look atsummary of the answersto surveyquestions

Survey question1:Are you male or female?

Survey question2:Would you prefer if the gym was unisex

65%of the females want the gym to be unisex, however only 20% of males want a unisex gym,So65%-20%=45% more women want the gym to be Unisex

Femalesspend more time on the cardio machine

This is obvious if you look atsummary of the answers to survey questions

Survey question1:Are you male or female?

Survey question3:How much time do you spend on the cardio machine?

Females spent an average of 38 minutes on thecardiomachine and males only spent an average of 14 minutes.Sofemales spend 38-14=24 minutes more on average

As time spent on cardio machine increases the time spend on the weight machine decreases

This is obvious if you look at scatterplot that showsanswers to survey questions

Survey question3:How much time do you spend on the cardio machine?
Survey question4:How much time do you spend on the weight machine?

As the time spent on the cardio machine increases the time spent on the weight decreases

Confidence intervals

Confidence intervalsfor

The proportion of females that want a unisex gym

The proportion of males that want a unisex gym.

90% confidence interval for sample proportion for females

​
Just considering the females

n=60,

sample proportion= 39/60=0.65

standard errorof sample proportion= XXXXXXXXXX

90% confident the true proportion is between

XXXXXXXXXX* XXXXXXXXXX=0.55 and XXXXXXXXXX* XXXXXXXXXX=0.75

90% confidence interval for sample proportion for males

​

Just considering the males
n=60,

sample proportion= 8/40=0.2

standard errorof sample proportion= XXXXXXXXXX

90% confident the true proportion is between

XXXXXXXXXX* XXXXXXXXXX =0.095 and XXXXXXXXXX* XXXXXXXXXX=0.305

Hypothesis tests that support the main findings

Beforegiving the survey to the gym customerwe knewwe wantedtoanswer the question
“Do most of the femaleswanta unisex gym?”

Which gives us the null and alternative hypothesis

H0:p≤0.5(to find the test stat assume p=0.5)HA:p>0.5

Sowe use the test stat

​
n=60,

sample proportion= 39/60=0.65

=

The p-value is P(Z>2.324) =0.01

Checking the entire process using a webpage we get the same result.

Since the p-value is less than 0.05 we reject H0because there is strong evidence the proportion is above50% ,So most of the femaleswant a unisex gym.

Test of the hypothesis “Do most of the males prefer a unisex gym”

Before giving the survey to the gym customersit was clear that summary of the dataset based on the survey can be used to answer the question“Do most of the maleswanta unisex gym?”

Which gives us the null and alternative hypothesis

H0:p≤0.5(assume p=0.5 to find the test stat)HA:p>0.5

Souse the teststat

​
just considering the malesn=40,

sample proportion=8/40=0.2

=

The p-value is P(Z>-3.7947) = XXXXXXXXXX

Checking the entire process using a webpage we get the same result.

Since the p-value is less than 0.05 we do not reject H0There is not strong evidence most of the males want a unisex gym.

Hypothesis test comparing proportions

Beforegiving the survey to the customers, it is was clear that the summary of the dataset can be used toanswer to the question

“Is there a relationship betweenvariables ‘gender’ and ‘should the gym be unisex’ “
So the appropriate null and alternative hypothesis is

H0:p1= p2 HA:p1≠ p2

theappropriatetest stat formulais

where

since you assume H0 is true this becomes

Using the information from the sample =4.41

Verfifying the results with an online webpage

http://www.socscistatistics.com/tests/ztest/Default2.aspx

Hypothesis test comparingmeans

Beforegiving 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 betweenvariables ‘gender’ and ‘time on cardio machine’ “
So the appropriate null and alternative hypothesis is

H0:µ1= µ2 HA:µ1≠ µ2

The appropriate test statistic is

which has a t distributionwith df

Since you assume H0 is true to calculate the test stat, the test stat becomes

Using the results from the surveys

=94 test stat=8.08

P-value0.000

Verifying the results with the webpage

https://www.graphpad.com/quickcalcs/ttest1/?Format=SD

data entry

Computer output

Reject H0there is strong evidence there is a difference between means

Before getting thesurveyyou can ask the questions

Is there a relationship betweenvariables.

Answer:

The test stat is

Since you assume the null hypothesis is true the test stat becomes

Test stat=

The formulas forandarevery complicated. (referto appendix3)

Test stat=

Information about women only gyms taken from the competitions webpages

“Why Women-Only Gyms Are More IdealForWomen Who Want To Stay Fit

Hitting up the gym could be one of the best solutions for women who want to commit to a healthier lifestyle. Since home exercise videos can be easily replaced by regular morning talk shows, it’s sometimes easier to just resort to a gym to stay on track. However, not a lot of women like the idea of losing weight while being surrounded by men. Many gyms cater to the needs of both men and women, but exclusive women’s gyms hold certain advantages, which could ultimately give women better results.

Comfortable and Appropriate

Women’s gymprovidethe ideal space for workouts, without the disturbance of working out around men. The gym is used as a place for doing workouts, and not as a dating venue. So, if you want to concentrate on doing your workouts in peace, a gym with female-only members is a much better option. Secondly, when you are not in shape, it is much more embarrassing to do exercises in front of men. When the gym has female-only members and trainers, you can do your workouts more comfortably without beingself-conscious, andgive exercising your all.

Modern society is an amazing mix of different cultures, and there are women coming from conservative backgrounds. Some women find it highly inappropriate to exercise in front of men, and it clashes with their religious beliefs. The women’s gym provides the ideal environment for workouts.

Customized to Specific Female Requirements

The programs and equipment available at an all-female gym are specially tailored for a woman’s physique and her specific needs. A woman’s body develops and works, in quite a different way than a male’s physique. Women’s gymhavespecialized equipment that provide women the opportunity to do workouts catering to their specific needs. Particularly the hydraulic equipment on the Curves circuit are specially designed, so that they conform to a woman’s body, which ensure the correct alignment during workouts.

The instructors are also more knowledgeable about a woman’s body, and specific exercises and workout plans, which are suitable for your body. A woman doing workouts doesn’t need to worry about over-straining, and the risk of injury is exponentially reduced.

Small and Intimate

Some co-ed gyms arehugeand one can feel lost in that environment, particularly women. A women’s gym is usually smaller-sized, and there is less likelihood of feeling out of place. Fewer members will also mean there is no time spent waiting for the use of exercise equipment, which not only saves time, but reduces a great amount of stress as well. You also tend to be more intimate with other members and form a closer relationship with the trainer.

Better Support

Only a woman can best understand the problems faced by another woman, whether it is physical or emotional. The women’s gym provides the best support group you can imagine. You are already struggling with being out of shape and overweight, and the right support can do wonders to your workout routines. You immediately start gainingconfidence, andknow that getting into shape is definitely within your reach. Other members of the gym would have gone through the same struggles you are facing, and you can always turn to them to clear your doubts and fears.

Considering the above points, women’s gyms aredefinitely moresuitable for women who want to get into shape or just stay fit. They provide the ideal environment for safe and peaceful workouts.”

Conclusion

There is strong evidencemostfemales want a Unisex gym and they concentrate on buying cardio machine.Most males are happy with a mixed gender gym.Soif Gym business XYZ buys a small new premises then it should make the gym only for females.

Appendix part 1 data set

Survey answers for 100 people

Appendixpart2notes on hypothesis testing

Hypothesis tests are used when you use a sample to answer a question about a population

An example of using a sample to answer a question about the population thenopinion polls famously predicted Clinton would win

If you are using a poll to answer the“yes or no “question “will Clinton win”, Before you even do the poll there are two possible predictions

1stpossible predicted answer to the question “Will Clinton Win” is “Yes, Clinton will win”
2ndpossible predicted answer to the question “Will Clinton Win” is “No, Clinton will not win”
Opinion polls are not perfect so there are two possible types of error
If the poll predicts “Yes” the real answer could be no
If the poll predicts “No” the real answer could be no

In statistics these would be defined as a type I and type II error these, many books and websites explain type I error and type II errorexplain Hypothesis testing will an even simpler example the pregnancy test.

The Woman is not pregnant is the null hypothesis H0
(Most of the time women are not pregnant, not being pregnant is the status quo)
Woman is pregnant is the alternative Hypothesis HA

If you were doing an opinion poll to check the claim“Will Clinton win”you make the alternative hypothesis “Clinton will win”HAand the null hypothesis “Clinton will not win”H0.

The biggestWolfever!

The biggestWolfever!

The biggestWolfever!

Come and save the greatest

child of all time!

Another way to remember this is by recalling the story of the Boy Who Cried Wolf.How shouldthevillagersplan for the futurebased on the story?

Deciding to believe something that can be wrong is risky.

If you hear a boy cry wolf you have two choices

Choice 1: Believe the boy,Ifyou are wrong you have made a type Ierror
Choice 2: Don’t believe the boy, If you are wrong you have made a type IIerror

More formally

Null Hypothesis: There is no wolf.
Alternate Hypothesis: There is a wolf.

Villagers believing the boy when there was no wolf (Rejecting null hypothesis incorrectly): Type I Error
Villagers not believing the boy when thereactually wasa wolf (Rejecting alternate hypothesis incorrectly): Type II Error.

Relating the example above to statistics, before obtaining a dataset you should realise ifyoudataset is a sample then there is a chance that a comment based on the sample does not apply to the population, the sample might “lie”, so you decide to believe the sample if the p-value is low.

Note that you can describe the computer output of hypothesis testing without referring to type I error, type II error, the null hypothesis and the alternative hypothesis

https://statistics.laerd.com/spss-tutorials/independent-t-test-using-spss-statistics.php

Investigates the relationship between
The categorical variable:Methodused to reduce cholesterol,diet or exercise.
TheQuantitative variable: cholesterol

This table provides useful descriptive statistics for the two groups that you compared, including the mean and standard deviation.

This table provides the actual results from the independent t-test.

You can see that the group means are statistically significantly different because the value in the "Sig. (2-tailed)" row is less than 0.05. Looking at theGroup Statisticstable, we can see that those people who undertook the exercise trial had lower cholesterol levels at the end of the programme than those who underwent a calorie-controlled diet.

Based on the results above, you could report the results of the study as follows (N.B., this does not include the results from your assumptions tests or effect size calculations):

This study found that overweight, physically inactive male participants had statistically significantly lower cholesterol concentrations (5.80 ± 0.38 mmol/L) at the end of an exercise-training programme compared to after a calorie-controlled diet (6.15 ± 0.52 mmol/L),t(38) = 2.428,p= 0.020.

I

Appendix 3 regression formula

Letbe sample data from a bivariate normal population(technically we havewhereis the sample size and will use the notationfor). Then we have the following sample statistics:

(sample mean for)​(sample mean for)

(sample variance for)​(sample variance for)

We will also use the following “sums of squares”:

and

Note: Sometimes we refer toas.

We have the following relationships between the sample statistics and the sums of squares:

or

and

or

The sample covariance betweenandis defined by

The sample correlation coefficient is defined as

whereand. By plugging the formulas ofandinto the formula forwe can easily derive that

.

The least squares regression line for the data has the form

where

and