^{1}

^{*}

^{2}

^{1}

^{3}

Every couple would want to have balanced sex of babies and a given number of children. But in reality, most couples do not achieve it. Some end up bearing particular sex of baby, this has caused a lot of pressure and untold hardship for couples. In Nigeria, the preference for a son is so strong that any family that does not have a son does not belong. The reason for the preference is mainly for economic and continuation of family lineage. Inability to bear the desired sex of a baby makes couples bear more children than they could cater for. This has caused poverty, overpopulation and reduces life expectancy for many families in Nigeria. In this paper, we have developed a model that will help couples select the desired sex of their babies and avoid unwanted pregnancies. The method is easy to apply whether educated or not. The application of the method will help to reduce family-based violence due to imbalanced sex of babies.

In many developing countries, especially Nigeria, there is preference for male over female children. Non-bearing of male child on time or no bearing at all leads to large family size that the couples may not be able to adequately cater for. This leads to poverty, overpopulation, low life expectancy. Every family would like to have at least one male child, not because males care for their parents more but for economic purposes and to have a male that will continue the family (generation) when the parents have died. But it is on record that females care more for their parents. So many families are in disarray because they have no son. Some men who could have been content with one wife become polygamous because the first wife did not bear a son. Some have divorced because of the same reason. Families that have balanced children are happier than the ones that have single sex children. For this reason; we explore all available knowledge to save families from this trauma that breaks homes.

Preference for male child is not only in Nigeria. Sex selection in China is largely dependent on fetal abortion and most of the aborted fetuses are females. The sex of the baby is determined through ultrasound and once it is a female child, especially at the second birth, the couple aborts the fetus. If a female child is conceived at the first birth, it will not be aborted because the couples still have chances of giving birth to a boy child. The reason is that there is a preference for male child to the female child [

It was observed that the vagina, cervix, and fallopian tubes undergo some changes in response to cyclical changes in levels of circulating ovarian hormones (estrogens and progesterone). Under the effect of estrogens, the cervical mucus turns from a thick, scant, highly viscous material in the pre-ovulation phase to a clear, watery, and abundant discharge that possesses the typical characteristic of being stretchy without breaking immediately before and after ovulation [

But in vitro studies have failed to demonstrate any differential action of different pH concentrations of Tyroides solution on the proportion of X and Y sperm, see [

In this paper, we have developed an estimator that determines the female fertile days: from where we determine; the day for male conception, the day for female conception and the safe period to avoid conception. We developed multivariate analysis model for the problem and test the hypothesis for the developed estimator. We stated and proof proposition backing the estimator and finally make inference about the determination of sex of a baby. The paper is divided into five sections. Section one above gives the general introduction of the work. Section two is devoted to literature review. Section three takes care of materials and methods. Section four takes care of data presentation and analysis while section five treats results and discussions.

The hallmark of conception is at the period of ovulation; some researchers [

Another researcher [

In this paper, we adopt median interval approach to determine the time of ovulation. The earlier study which centered on day 14 of menstrual cycle as ovulation day was deceptive because the menstrual length differs from one woman to the other. Some women have short and others have long menstrual cycle. But from theoretical and empirical studies, it was observed that ovulation takes place approximately two weeks from the start of the menstrual cycle. Hence, a woman with 28 days menstrual cycle is expected to ovulate on the 14^{th} day of her menstrual cycle. Note that 14^{th} day of menstrual cycle is not the same as 14^{th} day of the month. This implies that the median of the menstrual cycle with length 28 days is 14^{th} day. Since it is not certain that ovulation will occur exactly at the median but could vary because of some changes in the body, we develop a median interval estimate with 95% confidence that ovulation will occur within a given period. This period is divided into the lower and upper bound (L,U). The lower bound is used for the selection of female child and the upper bound if used for the selection of the male child. The idea is that the deposition of sperm days before the release of ovum (ova) will result to female child and the deposition of sperm close and after the release of ovum (ova) will result to male child. From the literatures in this field, it was known that Y-carrying chromosome is faster and have shorter life span than the X-carrying chromosome. If the sperm is deposited some days earlier in the female genitals before the release of the ovum (ova), the tendency is that the Y-carrying sperms must have died or become inactive to fertilize the ovum (ova), while the X-carrying sperm which are slower with longer life span will then fertilize the ovum (ova). Hence, if Y-carrying sperm first meet the egg (ovum) that has a constant X, we have XY, which will develop to a male child. But if X-carrying sperm first meet the egg, we have XX, which will develop to a female child. From the study of the natural method of family planning, it was observed that menstrual cycle and ovulation timing determines the sex of a baby. Our method is so easy and does not require education. Anybody, whether educated or not can apply it successfully and obtain the desired result. All that is required is to chart the menstrual cycle for at least three consecutive time (months) and determine the cycle length. We demonstrate this simple method here for easy understanding.

The type of data suitable for this kind of research is a secondary data because we want to deal with reality. We collected menstrual cycle charts of some selected women over a period of six months.

We developed a median interval estimator for the study. Our interest is to estimate the exact day of ovulation and if that is determined, then we can be sure that conception will take place at that day. From the motility of spermatozoa, we have stated that Y-carrying sperm moves faster but dies shortly, while the X-carrying sperm is sluggish and last longer. The problem now is to pin-point the day the ovum is released. Since we cannot be 100% sure of the day of ovulation due to variation in the human body, we need an estimator that will detect up to 95% the day of ovulation. Hence we constructed a median interval estimator at 5% level of significance. This gives us 95% confidence that ovulation will occur within the period. This period was divided into two regions, the lower and upper region (bound).

We also developed multivariate analysis model for the problem, since each individual’s menstrual cycles (samples) was collected for six variables (months). Each of the samples has six characteristics (variates) that need to be analyzed at the same time and their effects compared simultaneously; therefore, multivariate analysis provides the best model for such a problem. The multivariate analysis here is basically for inference, while the median interval estimator is the statistic to be tested. We determined the Hotelling’s T^{2} statistic; Mahalanobi’s D^{2} statistic and F-distribution for the problem. We test the null hypothesis that there are no significant differences among the median of the respective months and between the median of the first and the second samples (that is, ovulation occurs at the median of the menstrual cycle) against the alternative hypothesis which states the contrary (that is, ovulation occur close to the median). As we have stated, some circumstances due deviate ovulation from the centre; therefore, it can occur at the median (centre) or near the median, hence the need for interval estimator. An estimator is a statistic or a random variable that is used to determine the location of a parameter [

T 2 = n 1 n 2 n 1 + n 2 D 2 (1)

where T^{2} is the Hotelling’s T^{2} distribution; n_{1} is the sample data from the first sample; n_{2} is the sample data from the second sample respectively, and D^{2} is the Mahalanobi’s D^{2} statistic. Since the samples are from normal population, the mean and median coincides; hence, we can use both interchangeably.

D 2 = ( X ¯ 1 − X ¯ 2 ) T S − 1 ( X ¯ 1 − X ¯ 2 ) (2)

where ( X ¯ 1 − X ¯ 2 ) T is the difference in the mean vector of the sample mean vectors; the superscript T denote the transpose of the difference in mean vectors; and S − 1 denote the inverse of dispersion matrix [

F c a l = ( n 1 + n 2 − p − 1 ) p ( n 1 + n 2 − 2 ) ⋅ T 2 (3)

where F_{cal} denote F-calculated and p denote the parameters being estimated.

F t a b = F ( α ) ; p , ( n 1 + n 2 − p − 1 ) (4)

where F_{tab} denotes F-tabulated, this is the value of F as it is stated in the standard F statistical tables and ( α ) denotes the level of significance under which we make our inference. “VS” denotes versus. In this study, we test the hypothesis at 5% level of significance.

We determine the correlation between different menstrual cycles and make our inference.

In trying to achieve the above, we state some hypothesis that should be tested as follows:

1) Hypothesis:

Ho: μ 1 = μ 2 VS H_{1}: μ 1 ≠ μ 2 .

Ho: denote the null hypothesis which states that the (mean) median vector of sample 1 is the same as sample 2, this implies that ovulation occur at the median of menstrual cycle of every healthy woman in each sample, while H_{1}: stand for the alternative hypothesis which_{ }states the contrary (that is, the ovulation occurs close to the median).

2) Decision rule:

Accept Ho: if F_{cal} < F_{tab} and reject if otherwise.

The above statement means that if the calculated value of F-distribution is less than the tabulated value of the F-distribution, we shall accept the null hypothesis (Ho:) and if the contrary, we accept the alternative hypothesis (H_{1}:) at 5% level of significance.

3) Conclusion:

Since F_{cal} > F_{tab} (F_{cal} < F_{tab}), we reject Ho (accept Ho) and conclude that ovulation occurs close to the median (at the median). That is, there is significant difference between the two sample mean (median) vectors when ovulation occurs close to the median and there is no significant difference between the two sample mean (median) when ovulation occur at the median.

Conception of male child occurs at the upper bound (U) and conception of the female child occurs at the lower bound (L) and the boundary between L and U are the fertile days. Within this interval, a woman will see some signs that tell her that she is fertile and can conceive should she have unprotected sexual intercourse.

( n + 1 2 − K ≤ θ ≤ n + 1 2 + K ) = ( L ≤ θ ≤ U )

Proof:

Since we are not certain that the point estimator (median) will coincide with the population parameter (the exact day of ovulation), we replace it with interval estimator, that is, with the intervals for which we can assert with a reasonable degree of certainty that they will contain the parameter (the day of ovulation) under consideration.

Suppose that we have a large random sample, n ≥ 30 , from a population with the unknown mean μ and unknown variance σ 2 , where n is the sample size.

Then we have

Pr ( − Z α / 2 < x ¯ − μ σ n < Z α / 2 ) = 1 − α

Pr ( − Z α / 2 ∗ σ n < x ¯ − μ < Z α / 2 ∗ σ n ) = 1 − α

Pr ( μ − Z α / 2 ∗ σ n < x ¯ < μ + Z α / 2 ∗ σ n ) = 1 − α

We estimate the population standard deviation, σ , with the sample standard deviation, S, and let θ = x ¯ . Since our point estimator is the median, we have

Pr ( n + 1 2 − Z α / 2 ∗ S n < θ < n + 1 2 + Z α / 2 ∗ S n ) = 1 − α (5)

Equation (5) is the 95% confidence interval for θ that ovulation lies between the period (L, U). Where Pr stand for probability; n + 1 2 stand for the median; Z α / 2 stand for Z-value for two tail test from the standard normal distribution; θ stand for ovulation day; n + 1 2 − Z α / 2 ∗ S n stand for lower bound (L); n + 1 2 + Z α / 2 ∗ S n stand for upper bound (U) and 1 − α stand for confidence interval.

The raw data for this work are presented in Appendixs 1 and 2 respectively. In

Sample 1 (n_{1}) | ||||||
---|---|---|---|---|---|---|

S/N | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} |

1 | 24 | 24 | 25 | 26 | 28 | 28 |

2 | 25 | 25 | 26 | 27 | 29 | 29 |

3 | 26 | 26 | 27 | 28 | 30 | 30 |

4 | 27 | 27 | 28 | 1 | 31 | 1 |

5 | 28 | 28 | 29 | 2 | 1 | 2 |

6 | 29 | 29 | 30 | 3 | 2 | 3 |

7 | 30 | 30 | 31 | 4 | 3 | 4 |

8 | 1 | 31 | 1 | 5 | 4 | 5 |

9 | 2 | 1 | 2 | 6 | 5 | 6 |

10 | 3 | 2 | 3 | 7 | 6 | 7 |

11 | 4 | 3 | 4 | 8 | 7 | 8 |

12 | 5 | 4 | 5 | 9 | 8 | 9 |

13 | 6 | 5 | 6 | 10 | 9 | 10 |

14 | 7 | 6 | 7 | 11 | 10 | 11 |

15 | 8 | 7 | 8 | 12 | 11 | 12 |

16 | 9 | 8 | 9 | 13 | 12 | 13 |

17 | 10 | 9 | 10 | 14 | 13 | 14 |

18 | 11 | 10 | 11 | 15 | 14 | 15 |

19 | 12 | 11 | 12 | 16 | 15 | 16 |

20 | 13 | 12 | 13 | 17 | 16 | 17 |

21 | 14 | 13 | 14 | 18 | 17 | 18 |

22 | 15 | 14 | 15 | 19 | 18 | 19 |

23 | 16 | 15 | 16 | 20 | 19 | 20 |

24 | 17 | 16 | 17 | 21 | 20 | 21 |

25 | 18 | 17 | 18 | 22 | 21 | 22 |

26 | 19 | 18 | 19 | 23 | 22 | 23 |

27 | 20 | 19 | 20 | 24 | 23 | 24 |

28 | 21 | 20 | 21 | 25 | 24 | 25 |

29 | 22 | 21 | 22 | 26 | 25 | 26 |

30 | 23 | 22 | 23 | 27 | 26 | 27 |

31 | 24 | 23 | 24 | 28 | 27 | 28 |

32 | 24 | 25 | 28 | |||

33 | 25 | 26 |

Sample 2 (n_{2}) | ||||||
---|---|---|---|---|---|---|

S/N | X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} |

1 | 2 | 3 | 1 | 1 | 28 | 27 |

2 | 3 | 4 | 2 | 2 | 29 | 28 |

3 | 4 | 5 | 3 | 3 | 30 | 29 |

4 | 5 | 6 | 4 | 4 | 1 | 30 |

5 | 6 | 7 | 5 | 5 | 2 | 31 |

6 | 7 | 8 | 6 | 6 | 3 | 1 |

7 | 8 | 9 | 7 | 7 | 4 | 2 |

8 | 9 | 10 | 8 | 8 | 5 | 3 |

9 | 10 | 11 | 9 | 9 | 6 | 4 |

10 | 11 | 12 | 10 | 10 | 7 | 5 |

11 | 12 | 13 | 11 | 11 | 8 | 6 |

12 | 13 | 14 | 12 | 12 | 9 | 7 |

13 | 14 | 15 | 13 | 13 | 10 | 8 |

14 | 15 | 16 | 14 | 14 | 11 | 9 |

15 | 16 | 17 | 15 | 15 | 12 | 10 |

16 | 17 | 18 | 16 | 16 | 13 | 11 |

17 | 18 | 19 | 17 | 17 | 14 | 12 |

18 | 19 | 20 | 18 | 18 | 15 | 13 |

19 | 20 | 21 | 19 | 19 | 16 | 14 |

20 | 21 | 22 | 20 | 20 | 17 | 15 |

21 | 22 | 23 | 21 | 21 | 18 | 16 |

22 | 23 | 24 | 22 | 22 | 19 | 17 |

23 | 24 | 25 | 23 | 23 | 20 | 18 |

24 | 25 | 26 | 24 | 24 | 21 | 19 |

25 | 26 | 27 | 25 | 25 | 22 | 20 |

26 | 27 | 28 | 26 | 26 | 23 | 21 |

27 | 28 | 29 | 27 | 27 | 24 | 22 |

28 | 29 | 30 | 28 | 28 | 25 | 23 |

29 | 30 | 31 | 29 | 26 | 24 | |

30 | 1 | 1 | 30 | 27 | ||

31 | 2 | 1 | ||||

32 | 3 |

Layout of dispersion matrix for individual samples

( n i − 1 ) S i 2 = [ X 11 X 12 X 13 X 14 X 15 X 16 X 21 X 22 X 23 X 24 X 25 X 26 X 31 X 32 X 33 X 34 X 35 X 36 X 41 X 42 X 43 X 44 X 45 X 46 X 51 X 52 X 53 X 54 X 55 X 56 X 61 X 62 X 63 X 64 X 65 X 66 ]

where the variances are:

X 11 = ∑ 1 31 X 1 2 − n X ¯ 1 2 ; X 22 = ∑ 1 33 X 2 2 − n X ¯ 2 2 ; X 33 = ∑ 1 33 X 3 2 − n X ¯ 3 2 ; X 44 = ∑ 1 31 X 4 2 − n X ¯ 4 2 ; X 55 = ∑ 1 32 X 5 2 − n X ¯ 5 2 ; X 66 = ∑ 1 31 X 6 2 − n X ¯ 6 2

and covariance are:

X 12 = ∑ X 1 = 1 31 ∑ X 2 = 1 33 X 1 X 2 − n X ¯ 1 X ¯ 2 ; X 13 = ∑ X 1 = 1 31 ∑ X 3 = 1 33 X 1 X 3 − n X ¯ 1 X ¯ 3 ;

X 14 = ∑ X 1 = 1 31 ∑ X 4 = 1 31 X 1 X 4 − n X ¯ 1 X ¯ 4 ; X 15 = ∑ X 1 = 1 31 ∑ X 5 = 1 32 X 1 X 5 − n X ¯ 1 X ¯ 5 ;

X 16 = ∑ X 1 = 1 31 ∑ X 6 = 1 31 X 1 X 5 − n X ¯ 1 X ¯ 5 ; X 23 = ∑ X 2 = 1 33 ∑ X 3 = 1 33 X 2 X 3 − n X ¯ 2 X ¯ 3 ;

X 24 = ∑ X 2 = 1 33 ∑ X 4 = 1 31 X 2 X 4 − n X ¯ 2 X ¯ 4 ; X 25 = ∑ X 2 = 1 33 ∑ X 5 = 1 32 X 2 X 5 − n X ¯ 2 X ¯ 5 ;

X 26 = ∑ X 2 = 1 33 ∑ X 6 = 1 31 X 2 X 6 − n X ¯ 2 X ¯ 6 ; X 34 = ∑ X 3 = 1 33 ∑ X 4 = 1 31 X 3 X 4 − n X ¯ 3 X ¯ 4 ;

X 35 = ∑ X 3 = 1 33 ∑ X 5 = 1 32 X 3 X 5 − n X ¯ 3 X ¯ 5 ; X 36 = ∑ X 3 = 1 33 ∑ X 6 = 1 31 X 3 X 6 − n X ¯ 3 X ¯ 6 ;

X 45 = ∑ X 4 = 1 31 ∑ X 5 = 1 32 X 4 X 5 − n X ¯ 4 X ¯ 5 ; X 46 = ∑ X = 4 1 31 ∑ X 6 = 1 31 X 4 X 6 − n X ¯ 4 X ¯ 6 ;

X 56 = ∑ X 5 = 1 32 ∑ X 6 = 1 31 X 5 X 6 − n X ¯ 5 X ¯ 6

and symmetric entries of the matrix are

X 21 = X 12 ; X 31 = X 13 ; X 32 = X 23 ; X 41 = X 14 ; X 42 = X 24 ; X 43 = X 34 ; X 51 = X 15 ; X 52 = X 25 ; X 53 = X 35 ; X 54 = X 45 ; X 61 = X 16 ; X 62 = X 26 ; X 63 = X 36 ; X 64 = X 46 ; X 65 = X 56 .

Mean vectors from the first and second samples (n_{1} and n_{2}) are:

X ¯ ( 1 ) = [ X ¯ 1 X ¯ 2 X ¯ 3 X ¯ 4 X ¯ 5 X ¯ 6 ] = [ 15.7742 16.5152 16.5758 15.7097 16.3750 15.9032 ]

X ¯ ( 2 ) = [ X ¯ 1 X ¯ 2 X ¯ 3 X ¯ 4 X ¯ 5 X ¯ 6 ] = [ 14.6875 16.4667 15.0323 14.5000 15.5000 15.3448 ]

X ¯ ( 1 ) − X ¯ ( 2 ) = [ 1.0867 0.0485 1.5435 1.2097 0.8750 0.5584 ]

The dispersion (variance-covariance) matrix from the first sample (n_{1}) is:

( n 1 − 1 ) S 1 2 = [ 2317.4131 − 7305.0926 − 6786.0395 − 5714.0465 − 5595.5295 − 4819.6680 2616.1896 − 7941.8375 − 6723.3628 − 6686.1830 − 5808.6249 2650.0142 − 7298.8270 − 7263.4336 − 6384.4644 2252.3651 − 7092.2596 − 6268.8695 2619.5000 − 7186.0694 2398.7351 ]

The dispersion (variance-covariance) matrix from the second sample (n_{2}) is:

( n 2 − 1 ) S 2 2 = [ 2564.8750 − 6533.4943 − 5524.7876 − 4553.0625 − 4756.3438 − 4353.3676 2277.4337 − 6773.1315 − 5559.2474 − 5827.0155 − 5179.0074 2450.9287 − 5558.0663 − 5769.5198 − 5138.0291 1827.0000 − 5644.7500 − 5052.2386 2247.5000 − 6106.4098 2286.5763 ]

The pooled sample is:

( n 1 + n 2 − 2 ) S = [ 4882.2881 − 13838.5869 − 12310.8271 − 10267.1090 − 10351.8733 − 9173.0356 4893.6233 − 9714.9690 − 12282.6102 − 12513.1985 − 10987.6323 5100.9429 − 12856.8933 − 13032.9534 − 11522.4935 4079.3651 − 12737.0096 − 11321.1081 4867 − 13292.4792 4685.3114 ]

The dispersion (variance-covariance) matrix for the two samples is:

S = [ 81.3715 − 230.6431 − 205.1805 − 171.1185 − 172.5312 − 152.8839 81.5604 − 161.9162 − 204.7102 − 208.5533 − 183.1272 85.0157 − 214.2816 − 217.2159 − 192.0416 72.9894 − 212.2835 − 188.6851 81.1167 − 221.5413 78.0885 ]

The inverse of the dispersion matrix is:

S − 1 = [ 0.003312 − 0.00017 − 0.00063 − 0.00104 − 0.00114 − 0.00122 0.002928 − 0.00124 − 0.00081 − 0.00087 − 0.00095 0.002814 − 0.00064 − 0.00069 − 0.00074 0.002841 − 0.00076 − 0.0008 0.002598 − 0.00045 0.003172 ]

The Hotelling’s T^{2} statistic is:

T 2 = n 1 n 2 n 1 + n 2 ( x ¯ ( 1 ) − x ¯ 2 ) 1 S − 1 ( x ¯ ( 1 ) − x ¯ ( 2 ) )

T 2 = 32 × 30 62 ( 42.52789 ) = 658.4964

The calculated F statistic is:

F c a l = n 1 + n 2 − p − 1 p ( n 1 + n 2 − 2 ) ⋅ T 2

F c a l = 32 + 30 − 6 − 1 6 ( 32 + 30 − 2 ) × 42.52789 = 6.4973

The Tabulated F statistic at 5% level of significance is:

F t a b = F 6 , 60 ( 0.05 ) = F 6 , 60 ( 0.05 ) = 2.2500

1) Hypothesis

Ho: X ¯ n 1 = X ¯ n 2 Vs H_{1}: X ¯ n 1 ≠ X ¯ n 2 .

2) Conclusion

Since F_{cal} > F_{tab} (6.4973 > 2.2500), we reject Ho: and accept H_{1}: and conclude at 5% level of significance or with 95% confidence that ovulation occur close to the median in a menstrual cycle.

3) Choice of Sex for a Baby

θ = n + 1 2 ± Z α / 2 ∗ S n

θ = n + 1 2 ± Z 0.05 2 ∗ σ n

[ n + 1 2 − 1.96 σ n < θ < n + 1 2 + 1.96 σ n ] = [ L < θ < U ]

We substituted sample standard deviation (S) for population standard deviation ( σ ) to avoid confusing sample standard deviation (S) with dispersion matrix (S).

A couples who wish to select a female child should have intercourse on [ L < θ ] = n + 1 2 − 1.96 σ n days of the menstrual cycle.

And a couples who wish to select a male child should have intercourse on [ θ < U ] = n + 1 2 + 1.96 σ n days of the menstrual cycle.

Intercourse on [ θ ] day has equal chances of conceiving either of the sexes depending on whether ovulation has occurred or not. If ovulation has occurred on that day before intercourse, there is a high probability that a male child will be conceived but if sexual intercourse has occurred before ovulation, there is high probability that a female child will be conceived. In other words, a couple who wish to select a male child should have intercourse at the upper bound (U), and a couple who wish to select a female child should have intercourse at the lower bound (L).

4) Correlation Test

Correlation ( ρ i j ) = Covariance ( X i , X j ) Variace ( X i ) ⋅ Variance ( X j )

ρ 1 , 2 = − 0.00017 0.003312 × 0.002928 = − 0.05459

ρ 1 , 3 = − 0.00063 0.003312 × 0.002814 = − 0.20598

ρ 3 , 4 = − 0.00064 0.002814 × 0.002841 = − 0.22635

ρ 4 , 5 = − 0.00076 0.002841 × 0.002598 = − 0.27974

ρ 5 , 6 = − 0.00045 0.002598 × 0.003172 = − 0.15676

5) Correlation Results

There is weak relationship between the respective months under study. Each month relate with the preceding month, that is, one month cycle enters into another month to complete its cycle, etc. Hence, each cycle is not independent of the other and their relationship is in the opposite direction.

From the analysis we carried out in this work, we discovered with 95% confidence that ovulation occurs close to the median of the menstrual cycle. We were able to determine the interval estimator ( θ ) that estimates the range of fertile days. We determine the choice of sex for a baby at both lower and upper bound. Intercourse at the lower bound results in a female child with 95% confidence while intercourse at the upper bound results in a male child with 95% confidence. Intercourse on the day of ovulation has equal chances of getting either of the sexes depending on the timing of the intercourse, bearing in mind that Y-carrying sperm swim faster and die earlier. We observed that one menstrual cycle is dependent on the other cycle and they have weak inverse relationship.

With the model developed in this study, we determined that a female child can be conceived on the interval, [ L < θ ] = n + 1 2 − 1.96 σ n with probability very close to one and a male child can be conceived on the interval, [ θ < U ] = n + 1 2 + 1.96 σ n with probability very close to one. For women who want to avoid pregnancy, they should abstain from sexual intercourse on the interval [ n + 1 2 − 1.96 σ n < θ < n + 1 2 + 1.96 σ n ] , this interval represents the fertile days, and any intercourse within this period (L, U) will result in pregnancy. In order not to make a mistake, a couple looking for a boy child should have intercourse at midnight of the determined day of ovulation ( θ ) or within 24 hours from ( θ ). Our method is simple and should be followed carefully. First, determine the menstrual cycle length and determine the middle of it, which becomes the median. Secondly, determine the standard deviation of the cycle length. Thirdly, substitute the values in either (L) or (U) to determine the lower and upper bound. Our method has been tried on some selected women and the success rate was very impressive. If a couple was able to bear children with the desired balanced sex ratio, it will prevent the quest for more children. Many families bear more children when they are unable to get given sex of a child. With strict adherence to our model, family can avoid bearing more children than they can cater for, thereby reducing overpopulation. Pre-determining the sex of a baby and mastery of when to achieve and prevent pregnancy, the number of children and proper spacing between one pregnancy and another is termed family planning.

To practically demonstrate our method using the variate X_{1} from sample n_{2}, we have the following parameters: median = 16; standard deviation is 9.1; L = 13 and U = 19. Hence, the estimator lies within this interval [ 13 < θ < 19 ]. This implies that intercourse from 13^{th} and close to 16^{th} day of the menstrual cycle has 95% probability of conceiving a female child. Again, intercourse on the 16^{th} day of the menstrual cycle has equal probability of conceiving either of the sexes. Finally, intercourse from the 17^{th} to 19^{th} day of the menstrual cycle has 95% probability of conceiving a male child. Therefore, with our model, natural family planning and control of overpopulation are made easy.

The authors declare no conflicts of interest regarding the publication of this paper.

Amuji, H.O., Aronu, C.N., Onukwube, O.G. and Igboanusi, C. (2021) Modelling of Pre-Sex Selection: An Effective Method for Family Planning and Control of Overpopulation. Open Journal of Optimization, 10, 29-46. https://doi.org/10.4236/ojop.2021.102003