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Rewrite the following question or task so that it makes sense. Correct any formatting, grammar, or spelling errors, and ensure that mathematical expressions are properly presented using LaTeX. Do not change or remove [tex] [/tex] tags.

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Expand the following expressions:

1. (a) [tex]$\sin 20^\circ + \cos 80^\circ + 20\sin 43^\circ$[/tex]
(b) [tex]$\sin 135^\circ + \cos 49^\circ + \sqrt{2} \log ds^\circ$[/tex]

2. (a) [tex]$ \cos 0 + \sin \theta + \sin 90^\circ$[/tex]
(b) [tex]$\sin 45^\circ \cot 10^\circ \cos 45^\circ \sin 20^\circ$[/tex]

3. Prove that:
(a) [tex]$\sin 210^\circ = \sin 10^\circ$[/tex]
(b) [tex]$\sin 21^\circ = 2 \sin 10^\circ \cos 10^\circ$[/tex]

4. Verify the following identities:
(a) [tex]$4(\sin^4 A + \cos^4 A) = 1 + 2(\sin^3 A - \sin A)$[/tex]
(b) [tex]$\sin 3(1 - 1 \sin 1) = 1 \tan^2 0$[/tex]
(c) [tex]$\cos 10^\circ + \cos(0) = \sqrt{2} + 1$[/tex]
(d) [tex]$1 + \tan(20) + \sin 31 \theta^2 = 2$[/tex]

5. Solve for [tex]$\theta$[/tex]:
(a) [tex]$\sin 2\theta = 2 \sin \theta \cos \theta$[/tex]
(b) [tex]$\cos^3 0 + 1 = \sin 2 \theta$[/tex]



Answer :

GinnyAnswer
Let's start by analyzing the question and breaking down each part into meaningful and accurate steps to derive the result.

Given data:
- Sample size ([tex]\( n \)[/tex]): 85
- Population mean ([tex]\( \mu \)[/tex]): 22
- Population standard deviation ([tex]\( \sigma \)[/tex]): 13
- Lower bound: 19
- Upper bound: 23

We need to find the probability that a sample mean falls between 19 and 23.

Step-by-step solution:

1. Find the Standard Error:

The standard error of the mean (SEM) is given by the formula:

[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \][/tex]

Plugging in the given values:

[tex]\[ \text{SEM} = \frac{13}{\sqrt{85}} \][/tex]

2. Calculate the Z-scores for the Lower and Upper Bounds:

The Z-score for a value [tex]\( X \)[/tex] in a normal distribution is calculated as:

[tex]\[ Z = \frac{X - \mu}{\text{SEM}} \][/tex]

- For the lower bound (19):

[tex]\[ Z_{\text{lower}} = \frac{19 - 22}{\frac{13}{\sqrt{85}}} \][/tex]

- For the upper bound (23):

[tex]\[ Z_{\text{upper}} = \frac{23 - 22}{\frac{13}{\sqrt{85}}} \][/tex]

3. Look up the Cumulative Distribution Function (CDF):

The normal distribution table (or a computational tool) provides the cumulative probability up to a given Z-score.

- Determine the cumulative probability for both Z-scores.

4. Probability Calculation:

The probability that the sample mean lies between 19 and 23 is the difference in the cumulative probabilities for the upper and lower bounds.

[tex]\[ P(19 < \bar{X} < 23) = \text{CDF}(Z_{\text{upper}}) - \text{CDF}(Z_{\text{lower}}) \][/tex]

5. Final Numerical Values:

By calculating the Z-scores explicitly:

[tex]\[ Z_{\text{lower}} \approx -2.1276 \][/tex]

[tex]\[ Z_{\text{upper}} \approx 0.7092 \][/tex]

- From standard normal CDF tables, or using computational tools:
- The CDF value at [tex]\( Z_{\text{lower}} \approx -2.1276 \)[/tex] is approximately 0.0167.
- The CDF value at [tex]\( Z_{\text{upper}} \approx 0.7092 \)[/tex] is approximately 0.7609.

Thus, the probability [tex]\( P \)[/tex] is given by:

[tex]\[ P(19 < \bar{X} < 23) = 0.7609 - 0.0167 = 0.7442 \][/tex]

So, the probability that the sample mean lies between 19 and 23 is approximately [tex]\(0.7442\)[/tex], or 74.42%.

This answers the question requiring the calculation within this specific range.

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