Answer:
The 90% confidence interval is  [tex] -0.3433<  \mu_1 - \mu_2 < 2.1433[/tex]
Step-by-step explanation:
From the question we are told that
  The sample mean for men is  [tex]\= x_1 = 64.5 \ years[/tex]
  The sample mean for women is  [tex]\= x_2 = 63.6 \ years[/tex]
   The sample size for men  is  [tex]n_1 = 35[/tex]
   The sample size for women is  [tex]n_2 = 39[/tex]
   The standard deviation for men is [tex]s_1 = 3.0[/tex]
    The standard deviation for women is  [tex]s_2 = 3.5[/tex]
From the question we are told the confidence level is  90% , hence the level of significance is  Â
   [tex]\alpha = (100 - 90 ) \%[/tex]
=> Â [tex]\alpha = 0.10[/tex]
Generally from the normal distribution table the critical value  of  [tex]\frac{\alpha }{2}[/tex] is Â
  [tex]Z_{\frac{\alpha }{2} } =  1.645 [/tex]
Generally the standard error is mathematically represented as
   [tex]SE = \sqrt{\frac{s_1^2 }{n_1} + \frac{s_2^2}{n_2} }[/tex]
=> Â [tex]SE = \sqrt{\frac{3^2 }{35} + \frac{3.5^2}{39} }[/tex]
=> Â [tex]SE = 0.7558[/tex]
Generally the margin of error is mathematically represented as
    [tex]E = Z_{\frac{\alpha }{2} } * SE[/tex]
=> Â Â [tex]E = 1.645 * 0.7558[/tex]
=> Â Â [tex]E = 1.2433[/tex]
Generally 90% confidence interval is mathematically represented as Â
   [tex](\= x_1 - \= x_2) -E <  \mu_1 - \mu_2 <  (\= x_1 - \= x_2) -E [/tex]
=> Â Â [tex](64.5 - 63.6) -1.2433< Â \mu_1 - \mu_2 <(64.5 - 63.6) +1.2433[/tex]
=> Â Â [tex] -0.3433< Â \mu_1 - \mu_2 < 2.1433[/tex]
