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UNO [17]
3 years ago
12

A sample of 81 tobacco smokers who recently completed a new smoking-cessation program were asked to rate the effectiveness of th

e program on a scale of 1 to 10, with 10 corresponding to "completely effective" and 1 corresponding to "completely ineffective". The average rating was 5.6 and the standard deviation was 4.6. Construct a 95% confidence interval for the mean score. 5.2 < μ < 6.0 0 < μ < 5.6 4.6 < μ < 6.6 5.1 < μ < 6.1
Mathematics
1 answer:
Kruka [31]3 years ago
4 0

Answer:

95% confidence interval for the true mean score is [4.6 , 6.6].

Step-by-step explanation:

We are given that a sample of 81 tobacco smokers who recently completed a new smoking-cessation program were asked to rate the effectiveness of the program on a scale of 1 to 10.

The average rating was 5.6 and the standard deviation was 4.6.

Firstly, the pivotal quantity for 95% confidence interval for the true mean is given by;

                      P.Q. =  \frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }  ~ t_n_-_1

where, \bar X = sample average rating = 5.6

            s = sample standard deviation = 4.6

            n = sample of tobacco smokers = 81

            \mu = population mean score

<em>Here for constructing 95% confidence interval we have used One-sample t test statistics as we don't know about population standard deviation.</em>

<u>So, 95% confidence interval for the population mean score, </u>\mu<u> is ;</u>

P(-1.993 < t_8_0 < 1.993) = 0.95  {As the critical value of t at 80 degree of

                                            freedom are -1.993 & 1.993 with P = 2.5%}  

P(-1.993 < \frac{\bar X-\mu}{\frac{s}{\sqrt{n} } } < 1.993) = 0.95

P( -1.993 \times {\frac{s}{\sqrt{n} } } < {\bar X-\mu} < 1.993 \times {\frac{s}{\sqrt{n} } } ) = 0.95

P( \bar X-1.993 \times {\frac{s}{\sqrt{n} } } < \mu < \bar X+1.993 \times {\frac{s}{\sqrt{n} } } ) = 0.95

<u>95% confidence interval for</u> \mu = [ \bar X-1.993 \times {\frac{s}{\sqrt{n} } } , \bar X+1.993 \times {\frac{s}{\sqrt{n} } } ]

                                            = [ 5.6-1.993 \times {\frac{4.6}{\sqrt{81} } } , 5.6+1.993 \times {\frac{4.6}{\sqrt{81} } } ]

                                            = [4.6 , 6.6]

Therefore, 95% confidence interval for the true mean score is [4.6 , 6.6].

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The interest rate is 6.992%, if a bank advertises that it compounds money quarterly and that it will take Double your money in 10 years.

Step-by-step explanation:

The given is,

                          Compounds money quarterly

                          Double your money in 10 years

Step:1

              Formula to calculate future investment with compounded quarterly,

                                                        A =P(1+\frac{r}{n} )^{nt}...............................(1)

              Where, A - Future amount

                           P - Initial investment\

                            r - Rate of interest

                            n - No. of compounding in a year

                             t - No. of years

Step:2

               Let, P = X

                      A = 2X ( Double your money )

              From given, n - 4 ( for compounding quarterly )

                                    t - 10 years

               From equation (1)

                                                        2X =X(1+\frac{r}{4} )^{(4)(10)}

                                                         \frac{2X}{X}  =(1+\frac{r}{4} )^{(4)(10)}

                                                            2  =(1+\frac{r}{4} )^{40}

          Take root 40^{th} root on both side,

                                                         \sqrt[40]{2} = (1+\frac{r}{4} )

                                            1.017479692 = (\frac{4+r}{4} )

                                     (1.017479692)(4) = (r+4)

                                                  4.06992=(r+4)

                                                              r = 4.06992 -4

                                                              r =( 0.06992)(100)

                                                              r = 6.992 %

Result:

             The interest rate is 6.992%, if a bank advertises that it compounds money quarterly and that it will take Double your money in 10 years.

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